List of publications and abstracts
[Books],
[Edited Volumes],
[Papers],
[Proceedings],
[Thesis].

The Defocusing Nonlinear Schrödinger Equation:
From Dark Solitons to Vortices and Vortex Rings.
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
SIAM, Philadelphia, 2015.
429 pages., 227 illus., Softcover, ISBN: 9781611973938
[Review],
[Website],
[Preface],
[Contents],
[Index],
[Cover].

Emergent Nonlinear Phenomena in BoseEinstein Condensates:
Theory and Experiment.
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Springer Series on Atomic, Optical, and Plasma Physics, Vol. 45, 2008.
406 p., 108 illus., Hardcover, ISBN: 9783540735908
[Website],
[Title page],
[Foreword],
[Preface],
[Contents],
[Contributors],
[Cover],
[Have a look inside!].



Issue on Localized Excitations in Nonlinear Complex Systems (LENCOS'09):
Editors:
R. CarreteroGonzález,
J. Cuevas,
D.J. Frantzeskakis,
P.G. Kevrekidis, and
F. Palmero.
Discrete and Continuous Dynamical Systems  Series S,
Volume: 4, Number: 5, October 2011.
[Website],
[Preface],
[Contents],
[Cover].

Localized Excitations in Nonlinear Complex Systems (LENCOS'12):
Current State of the Art and Future Perspectives
Editors:
R. CarreteroGonzález,
J. CuevasMaraver,
D.J. Frantzeskakis,
N. Karachalios,
P.G. Kevrekidis, and
F. Palmero.
Series: Nonlinear Systems and Complexity, Volume 7, 2014.
432 p., 117 illus. in color, ISBN: 9783319020570
[Website],
[Front Matter / Preface / Contents / Contributors].


 Stability of single and multiple matterwave dark solitons in collisionally
inhomogeneous BoseEinstein condensates.
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Submitted, August 2016.
Abstract. PDF.
 On the spontaneous timereversal symmetry breaking in synchronouslypumped passive Kerr resonators.
J. Rossi,
R. CarreteroGonzález.
P.G. Kevrekidis, and
M. Haragus.
Submitted, May 2016.
Abstract. PDF.
 Dark spherical shell solitons in threedimensional BoseEinstein condensates:
Existence, stability, and dynamics.
Wenlong Wang,
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. A 93 (2016) 023630.
Abstract. PDF.
Movie.
 Robust Vortex Lines, Vortex Rings and Hopfions in 3D BoseEinstein Condensates.
R.N. Bisset,
Wenlong Wang,
C. Ticknor,
R. CarreteroGonzález,
D.J. Frantzeskakis,
L.A. Collins, and
P.G. Kevrekidis.
Phys. Rev. A 92 (2015) 063611.
Abstract. PDF.
Movies.
 Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the GrossPitaevskii Equation.
J.C. Tzou,
P.G. Kevrekidis,
T. Kolokolnikov, and
R. CarreteroGonzález.
SIAM J. Appl. Dyn. Syst. 15 (2016) 904922.
Abstract. PDF.
 NonConservative Variational Approximation for Nonlinear Schrödinger Equations.
J. Rossi,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Submitted, Aug 2015.
Abstract. PDF.
 Generating and Manipulating Quantized Vortices OnDemand in a BoseEinstein Condensate: a Numerical Study.
B. Gertjerenken,
P.G. Kevrekidis,
R. CarreteroGonzález, and
B.P. Anderson,
Phys. Rev. A 93 (2016) 023604.
Abstract. PDF.
Movies.
 Bifurcation and Stability of Single and Multiple Vortex Rings
in ThreeDimensional BoseEinstein Condensates.
R.N. Bisset,
Wenlong Wang,
C. Ticknor,
R. CarreteroGonzález,
D.J. Frantzeskakis,
L.A. Collins, and
P.G. Kevrekidis.
Phys. Rev. A 92 (2015) 043601.
Abstract. PDF.
Selected for Phys. Rev. A's Kaleidoscope.
 Solitons Riding on Solitons and the Quantum Newton's Cradle.
Manjun Ma,
R. Navarro, and
R. CarreteroGonzález.
Phys. Rev. E 93 (2016) 022202.
Abstract. PDF.
 Stabilization of ring dark solitons in BoseEinstein condensates.
Wenlong Wang,
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
Tasso J. Kaper, and
Manjun Ma.
Phys. Rev. A 92 (2015) 033611.
Abstract. PDF.
 Proper Orthogonal Decomposition Methods for the Analysis of RealTime Data:
Exploring Peak Clustering in a Secondhand Smoke Exposure Intervention.
V. Berardi,
R. CarreteroGonzález,
N.E. Klepeis,
A. Palacios,
J. Bellettiere,
S. Hugues,
S. Obayashi, and
M.F. Hovell.
J. Comp. Sci. 11 (2015) 102111.
Abstract. PDF.
 Optoelectronic Chaos in a Simple Light Activated Feedback Circuit.
K.L. Joiner,
F. Palmero, and
R. CarreteroGonzález.
Int. J. Bifurcation and Chaos 26 (2016) 1650080.
Abstract. PDF.
 Vortex Nucleation in a Dissipative Variant of the Nonlinear Schrödinger Equation under Rotation.
R. CarreteroGonzález,
P.G. Kevrekidis, and
T. Kolokolnikov.
Physica D 317 (2016) 114.
Abstract. PDF.
Movies.
 Dynamics of vortex dipoles in anisotropic BoseEinstein condensates.
R.H. Goodman,
P.G. Kevrekidis, and
R. CarreteroGonzález.
SIAM J. Appl. Dyn. Syst. 14 (2015) 699729.
Abstract. PDF.
 Scattering and leapfrogging of vortex rings in a superfluid.
R.M. Caplan,
J.D. Talley,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Fluids 26 (2014) 097101.
Abstract. PDF.
 Dynamic and Energetic Stabilization of Persistent Currents in BoseEinstein Condensates.
K.J.H. Law,
T.W. Neely,
P.G. Kevrekidis,
B.P. Anderson,
A.S. Bradley, and
R. CarreteroGonzález.
Phys. Rev. A 89 (2014) 053606.
Abstract. PDF.
 A Tale of Two Distributions: From Few To Many Vortices In QuasiTwoDimensional BoseEinstein Condensates.
T. Kolokolnikov,
P.G. Kevrekidis, and
R. CarreteroGonzález.
Proc. R. Soc. A 470 (2014) 20140048.
Abstract. PDF.
 Exploring Vortex Dynamics in the Presence of Dissipation: Analytical and Numerical Results.
D. Yan,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P.G. Kevrekidis,
N.P. Proukakis, and
D. Spirn.
Phys. Rev. A 89 (2014) 043613.
Abstract. PDF.
 Nonlinear PTSymmetric models Bearing Exact Solutions.
H. Xu,
P.G. Kevrekidis,
Q. Zhou,
D.J. Frantzeskakis,
V. Achilleos, and
R. CarreteroGonzález.
Romanian J. Phys. 59 (2014) 185194.
Abstract. PDF.
 Directed Ratchet Transport in Granular Chains.
V. Berardi,
J. Lydon,
P.G. Kevrekidis,
C. Daraio, and
R. CarreteroGonzález.
Phys. Rev. E 88 (2013) 022912.
Abstract. PDF.
 From Nodeless Clouds and Vortices to Gray Ring Solitons and SymmetryBroken States in TwoDimensional Polariton Condensates.
A.S. Rodrigues,
P.G. Kevrekidis,
R. CarreteroGonzález,
J. CuevasMaraver,
D.J. Frantzeskakis, and
F. Palmero,
J. Phys.: Condens. Matter 26 (2014) 155801.
Abstract. PDF.
Movies.
 Exploring Rigidly Rotating Vortex Configurations and their Bifurcations in Atomic BoseEinstein Condensate.
A.V. Zampetaki,
R. CarreteroGonzález,
P.G. Kevrekidis,
F.K. Diakonos, and
D.J. Frantzeskakis.
Phys. Rev. E 88 (2013) 042914.
Abstract. PDF.
 PhaseShift Plateaus in the Sagnac Effect for Matter Waves.
M.C. Kandes,
R. CarreteroGonzález, and
M.W.J. Bromley.
Submitted, 2013.
Abstract. PDF.
 Dynamics of Few Corotating Vortices in BoseEinstein Condensates.
R. Navarro,
R. CarreteroGonzález,
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
M.W. Ray,
E. Altuntaş, and
D.S. Hall.
Phys. Rev. Lett. 110 (2013) 225301.
Abstract. PDF.
 Inelastic Collisions of Solitary Waves in Anisotropic BoseEinstein Condensates:
SlingShot Events and Expanding Collision Bubbles.
C. Becker,
K. Sengstock,
P. Schmelcher,
R. CarreteroGonzález, and
P.G. Kevrekidis.
New J. Phys. 15 (2013) 113028.
Abstract. PDF.
 Nonlinear localized modes in twodimensional electrical lattices.
L.Q. English,
F. Palmero,
J. Stormes,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. E 88 (2013) 022912.
Abstract. PDF.
 Solitons and their ghosts in PTsymmetric systems with defocusing
nonlinearities.
V. Achilleos,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity
7, (2014) 342.
Abstract. PDF.
 Symmetrybreaking Effects for Polariton Condensates in DoubleWell Potentials.
A.S. Rodrigues,
P.G. Kevrekidis,
J. Cuevas,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Progress in Optical Science and Photonics, 1 (2013) 509529.
Abstract. PDF.
 Characteristics of TwoDimensional Quantum Turbulence in a Compressible Superfluid.
T.W. Neely,
A.S. Bradley,
E.C. Samson,
S.J. Rooney,
E.M. Wright,
K.J.H. Law,
R. CarreteroGonzález,
P.G. Kevrekidis,
M.J. Davis, and
B.P. Anderson.
Phys. Rev. Lett. 111 (2013) 235301.
Abstract. PDF.
[Supplemental material].
[Movie].
 Dark solitons and vortices in PTsymmetric nonlinear media:
from spontaneous symmetry breaking to nonlinear PT phase transitions.
V. Achilleos,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Phys. Rev. A 86 (2012) 013808.
Abstract. PDF.
 Vortices in BoseEinstein Condensates: (Super)fluids with a twist.
P.G. Kevrekidis.
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Dynamical Systems Magazine, October 2011.
Abstract. WWW.
 A ModulusSquared Dirichlet Boundary Condition for
TimeDependent Complex Partial Differential Equations and
its Application to the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
SIAM J. Sci. Comput., 36 (2014) A1A19.
Abstract. PDF.
 A TwoStep HighOrder Compact Scheme for the Laplacian Operator and its Implementation in an
Explicit Method for Integrating the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
J. Comput. Appl. Math. 251 (2013) 3346.
Abstract. PDF.
 Numerical Stability of Explicit RungeKutta FiniteDifference Schemes for the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
App. Num. Math. 71 (2013) 2440.
Abstract. PDF.
Awarded the 6th most successful IMACS paper published in 2013 in Applied Numerical Mathematics.
 Dynamics of Vortex Dipoles in Confined BoseEinstein Condensates.
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P. Schmelcher, and
D.S. Hall.
Phys. Lett. A 375 (2011) 30443050.
Abstract. PDF.
 Generation of localized modes in an electrical lattice using subharmonic driving.
L.Q. English,
F. Palmero,
P. Candiani,
J. Cuevas,
R. CarreteroGonzález,
P.G. Kevrekidis, and
A.J. Sievers.
Phys. Rev. Lett. 108 (2012) 084101.
Abstract. PDF.
 Multiple darkbright solitons in atomic BoseEinstein condensates.
D. Yan,
J.J. Chang,
C. Hamner,
P.G. Kevrekidis.
P. Engels,
V. Achilleos,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
Phys. Rev. A 84 (2011) 053630.
Abstract. PDF.
 Discrete breathers in a nonlinear electric line: Modeling, Computation and Experiment.
F. Palmero,
L.Q. English,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. E 84 (2011) 026605.
Abstract. PDF.
 GuidingCenter Dynamics of Vortex Dipoles in BoseEinstein Condensates.
S. Middelkamp,
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P. Schmelcher,
D.V. Freilich, and
D.S. Hall.
Phys. Rev. A 84 (2011) 011605(R).
Abstract. PDF.
 Nonlinear Excitations, Stability Inversions and
Dissipative Dynamics in Quasionedimensional Polariton Condensates.
J. Cuevas,
A.S. Rodrigues,
R. CarreteroGonzález,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. B 83 (2011) 245140.
Abstract. PDF.
 Variational approximations in discrete nonlinear Schrödinger
equations with nextnearestneighbor couplings.
C. Chong,
R. CarreteroGonzález,
B.A. Malomed, and
P.G. Kevrekidis.
Physica D
240 (2011) 12051212.
Abstract. PDF.
 Controlling directed transport of matterwave solitons using the ratchet effect.
M.A. Rietmann,
R. CarreteroGonzález, and
R. Chacon.
Phys. Rev. A 83 (2011) 053617.
Abstract. PDF.
 Emergence and Stability of Vortex Clusters in BoseEinstein Condensates: a Bifurcation Approach near the Linear Limit.
S. Middelkamp,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
Physica D,
240 (2011) 14491459.
Abstract. PDF.
 Vortex Interaction Dynamics in Trapped BoseEinstein Condensates.
P.J. Torres,
R. CarreteroGonzález,
S. Middelkamp,
P. Schmelcher,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Comm. Pure Appl. Ana.
10 (2011) 15891615.
Abstract. PDF.
 Dynamics of DarkBright Solitons in CigarShaped BoseEinstein Condensates.
S. Middelkamp,
J.J. Chang,
C. Hamner,
R. CarreteroGonzález,
P.G. Kevrekidis,
V. Achilleos,
D.J. Frantzeskakis,
P. Schmelcher, and
P. Engels.
Phys. Lett. A
375 (2011) 642646.
Abstract. PDF.
Darkbright (DB) oscillation movies:
[ Movie#1, Fig. 3 ]:
Single DB for parameters in Nature Phys. 4, 496 (2008):
N_{D}=92,432 and N_{B}=7,973, (f_{z},f_{y},f_{x})=(85,133,5.9) Hz,
[ Movie#2, Fig. 4.a ]:
Single DB with bright soliton transverse dynamics:
N_{D}=88,181 and N_{B}=1,058, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.d ]:
Two interacting DBs with outofphase (attractive) bright solitons:
N_{D}=5,243 and N_{B}=817, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.e ]:
Two interacting DBs with inphase (repulsive) bright solitons:
N_{D}=5,331 and N_{B}=907, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz.
 Bifurcations, Stability and Dynamics of Multiple MatterWave Vortex States.
S. Middelkamp,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
Phys. Rev. A 82 (2010) 013646.
Abstract. PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 2, issue 8 (2010).
 Controlling the transverse instability of dark solitons and
nucleation of vortices by a potential barrier.
Manjun Ma,
R. CarreteroGonzález,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
B.A. Malomed.
Phys. Rev. A 82 (2010) 023621.
Abstract. PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 2, issue 9 (2010).
 Existence, Stability, and Scattering of Bright Vortices in the CubicQuintic Nonlinear Schrödinger Equation.
R.M. Caplan,
R. CarreteroGonzález.
P.G. Kevrekidis, and
B.A. Malomed.
Math. Comput. Simulat. 82 (2012) 11501171.
Abstract. PDF.
 Stability and dynamics of matterwave vortices in the
presence of collisional inhomogeneities and dissipative perturbations.
S. Middelkamp,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
J. Phys. B
43 (2010) 155303.
Abstract. PDF.
 Manipulation of Vortices by Localized Impurities in BoseEinstein Condensates.
M.C. Davis,
R. CarreteroGonzález,
Z. Shi,
K.J.H. Law,
P.G. Kevrekidis, and
B.P. Anderson.
Phys. Rev. A 80 (2009) 023604.
Abstract. PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 1, issue 3 (2009).
 Phase Separation and Dynamics of Twocomponent BoseEinstein Condensates.
R. Navarro,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. A 80 (2009) 023613.
Abstract. PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 1, issue 3 (2009).
 Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation.
R.M. Caplan,
Q.E. Hoq,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Optics. Comm. 282 (2009) 13991405.
Abstract. PDF.
 Spinor BoseEinstein condensate past an obstacle.
A.S. Rodrigues,
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P. Schmelcher,
T.J. Alexander, and.
Yu.S. Kivshar.
Phys. Rev. A 79 (2009) 043603.
Abstract. PDF.
 Dissipative Solitary Waves in Periodic Granular Crystals.
R. CarreteroGonzález,
D. Khatri,
M.A. Porter,
P.G. Kevrekidis, and
C. Daraio.
Phys. Rev. Lett. 102 (2009) 024102.
Abstract. PDF.
 Controlling chaos of a BoseEinstein condensate loaded into a moving
optical Fouriersynthesized lattice.
R. Chacon,
D. Bote, and
R. CarreteroGonzález.
Phys. Rev. E 78 (2008) 036215.
Abstract. PDF.
 A Map Approach to Stationary Solutions of the
Discrete Nonlinear Schrödinger Equation.
R. CarreteroGonzález.
Book chapter for: Discrete Nonlinear Schrödinger Equation:
Mathematical Analysis, Numerical Computations and Physical Perspectives,
P.G. Kevrekidis (Ed),
Springer Tracts in Modern Physics, Vol. 232, 2009.
Abstract. PDF.
 Structure and stability of twodimensional BoseEinstein condensates
under both harmonic and lattice confinement.
K.J.H. Law,
P.G. Kevrekidis,
B.P. Anderson,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
J. Phys. B,
41 (2008) 195303.
Abstract. PDF.
 Surface Solitons in Three Dimensions.
Q.E. Hoq,
R. CarreteroGonzález,
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis,
Yu.V. Bludov, and
V.V. Konotop.
Phys. Rev. E 78 (2008) 036605.
Abstract. PDF.
 Multistable Solitons in HigherDimensional CubicQuintic Nonlinear Schrödinger Lattices.
C. Chong,
R. CarreteroGonzález,
B.A. Malomed, and
P.G. Kevrekidis.
Physica D,
238 (2009) 126136.
Abstract. PDF.
 Nonlinear dynamics of Bosecondensed gases by means of a
qGaussian variational approach.
A.I. Nicolin and
R. CarreteroGonzález.
Physica A 387 (2008) 6032.
Abstract. PDF.
 Solitons in onedimensional nonlinear Schrödinger lattices with a local inhomogeneity.
F. Palmero,
R. CarreteroGonzález,
J. Cuevas,
P.G. Kevrekidis, and
W. Królikowski.
Phys. Rev. E 77 (2008) 036614.
Abstract. PDF.
 Dynamics of Vortex Formation in Merging BoseEinstein Condensate Fragments.
R. CarreteroGonzález,
B.P. Anderson,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
C.N. Weiler.
Phys. Rev. A 77 (2008) 033625.
Abstract. PDF.
 Resonant energy transfer in BoseEinstein condensates.
A.I. Nicolin,
M.H. Jensen,
J.W. Thomsen, and
R. CarreteroGonzález.
Physica D,
237 (2008) 24762481.
Abstract. PDF.
 Nonlinear Waves in BoseEinstein Condensates:
Physical Relevance and Mathematical Techniques.
R. CarreteroGonzález,
D.J. Frantzeskakis, and
P.G. Kevrekidis.
Nonlinearity 21 (2008) R139R202.
Abstract. PDF.
 Radially Symmetric Nonlinear States of Harmonically Trapped BoseEinstein Condensates.
G. Herring,
L.D. Carr,
R. CarreteroGonzález,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. A 77 (2008) 023625.
Abstract. PDF.
 Faraday waves in BoseEinstein condensates.
A.I. Nicolin,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. A 76 (2007) 063609.
Abstract. PDF.
 Extended Nonlinear Waves in Multidimensional Dynamical Lattices.
Q.E. Hoq,
J. Gagnon,
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Math. Comput. Simulat.,
80 (2009) 721731.
Abstract. PDF.
 Polarized States and Domain Walls in Spinor BoseEinstein Condensates.
H.E. Nistazakis,
D.J. Frantzeskakis.
P.G. Kevrekidis,
B.A. Malomed,
R. CarreteroGonzález, and
A.R. Bishop.
Phys. Rev. A 76 (2007) 063603.
Abstract. PDF.
 BrightDark Soliton Complexes in Spinor BoseEinstein Condensates.
H.E. Nistazakis,
D.J. Frantzeskakis.
P.G. Kevrekidis,
B.A. Malomed, and
R. CarreteroGonzález.
Phys. Rev. A 77 (2008) 033612.
Abstract. PDF.
 Symmetry breaking in linearly coupled dynamical lattices.
G. Herring,
P.G. Kevrekidis.
B.A. Malomed,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. E 76 (2007) 066606.
Abstract. PDF.
 NonEquilibrium Dynamics and Superfluid Ring Excitations in Binary BoseEinstein Condensates.
K.M. Mertes,
J. Merrill,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P.G. Kevrekidis, and
D.S. Hall.
Phys. Rev. Lett. 99 (2007) 190402.
Abstract. PDF.
Movies.
 Čerenkovlike radiation in a binary superfluid flow past an obstacle.
H. Susanto,
P.G. Kevrekidis.
R. CarreteroGonzález,
B.A. Malomed,
D.J. Frantzeskakis, and
A.R. Bishop.
Phys. Rev. A 75 (2007) 055601.
Abstract. PDF.
 Vortex Structures Formed by the Interference of Sliced Condensates.
R. CarreteroGonzález,
N. Whitaker,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. A,
77 (2008) 023605.
Abstract. PDF.
 Mode locking of a driven BoseEinstein condensate.
A.I. Nicolin,
M.H. Jensen, and
R. CarreteroGonzález.
Phys. Rev. E
75 (2007) 036208.
Abstract. PDF.
 Rotating matter waves in BoseEinstein condensates.
T. Kapitula,
P.G. Kevrekidis, and
R. CarreteroGonzález.
Physica D,
233 (2007) 112137.
Abstract. PDF.
 Discrete surface solitons in two dimensions.
H. Susanto,
P.G. Kevrekidis,
B.A. Malomed,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. E 75 (2007) 056605.
Abstract. PDF.
 Mobility of Discrete Solitons in Quadratic Nonlinear Media.
H. Susanto,
P.G. Kevrekidis,
R. CarreteroGonzález,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. Lett. 99 (2007) 214103.
Abstract. PDF.
 Skyrmionlike states in two and threedimensional dynamical lattices.
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
B.A. Malomed, and
F.K. Diakonos.
Phys. Rev. E
75 (2007) 026603.
Abstract. PDF.
 Multipolemode solitons in Bessel optical lattices.
Y.V. Kartashov,
R. CarreteroGonzález,
B.A. Malomed,
V.A. Vysloukh, and
Ll. Torner.
Optics Express
13, 26 (2006) 1070310710.
Abstract. PDF.
 Soliton trains and vortex streets as a form of Cerenkov radiation in
trapped BoseEinstein condensates.
R. CarreteroGonzález,
P.G. Kevrekidis,
D.J. Frantzeskakis.
B.A. Malomed,
S. Nandi, and
A.R. Bishop.
Math. Comput. Simulat.,
74 (2007) 361369.
Abstract. PDF.
 Dynamics and Manipulation of MatterWave Solitons in Optical Superlattices.
M.A. Porter,
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Lett. A
352 (2006) 210.
Abstract. PDF.
 Vector solitons with an embedded domain wall.
P.G. Kevrekidis,
H. Susanto,
R. CarreteroGonzález,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. E 72 (2005) 066604.
Abstract. PDF.
 Discrete Solitons and Vortices on Anisotropic Lattices.
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
B.A. Malomed,
and A.R. Bishop.
Phys. Rev. E 72 (2005) 046613.
Abstract. PDF.
 Multistable Solitons of the CubicQuintic Discrete Nonlinear Schrödinger Equation.
R. CarreteroGonzález,
J.D. Talley,
C. Chong, and
B.A. Malomed.
Physica D
216 (2006) 7789.
Abstract. PDF.
 Trapped bright solitons in the presence of localized inhomogeneities.
G. Herring,
P.G. Kevrekidis,
R. CarreteroGonzález,
B.A. Malomed,
D.J. Frantzeskakis,
and A.R. Bishop.
Phys. Lett. A 345 (2005) 144.
Abstract. PDF.
 ThreeDimensional Nonlinear Lattices:
From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes.
R. CarreteroGonzález,
P.G. Kevrekidis,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. Lett. 94 (2005) 203901.
Abstract. PDF.
 Multifrequency Synthesis Using two Coupled Nonlinear Oscillator Arrays.
A. Palacios,
R. CarreteroGonzález,
P. Longhini,
N. Renz,
V. In,
A. Kho,
B. Meadows, and
J. Neff.
Phys. Rev. E 72 (2005) 026211.
Abstract. PDF.
 Vortices in BoseEinstein Condensates: Some Recent Developments.
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis, and
I.G. Kevrekidis.
Mod. Phys. Lett. B,
18 (2004) 14811505.
Abstract. PDF.
 Nonlinear Lattice Dynamics of BoseEinstein Condensates.
M.A. Porter,
R. CarreteroGonzález,
P.G. Kevrekidis and
B.A. Malomed,
Chaos,
15 (2005) 015115.
Abstract. PDF.
Selected for the Virtual Journal of Biological Physics Research volume 9, issue 7 (2005).
 Statics, Dynamics and Manipulation of Bright Matterwave Solitons in Optical Lattices.
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
B.A. Malomed,
G. Herring and
A.R. Bishop.
Phys. Rev. A, 71 (2005) 023614.
Abstract. PDF.
 HigherOrder Vortices in Nonlinear Dynamical Lattices.
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Focus on Soliton Research,
Editor L.V. Chen, Nova
Science Publishers (2006) 139166.
Abstract. PDF.

Controlling the motion of dark solitons by means of periodic potentials:
Application to BoseEinstein condensates in optical lattices.
G. Theocharis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P.G. Kevrekidis and
B.A. Malomed.
Phys. Rev. E 71 (2005) 017602.
Abstract. PDF.
 Threedimensional solitary waves and vortices in a discrete
nonlinear Schrödinger lattice.
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis and
R. CarreteroGonzález.
Phys. Rev. Lett.
93 (2004) 080403.
Abstract. PDF.
 Domain Walls of SingleComponent BoseEinstein Condensates
in External Potentials.
P.G. Kevrekidis,
B.A. Malomed,
D.J.
Frantzeskakis,
A.R. Bishop,
H.E. Nistazakis and
R.
CarreteroGonzález.
Math. Comput. Simulat., 69 (2005) 334345.
Abstract. PDF.
 A ParrinelloRahman Approach to Vortex Lattices.
R. CarreteroGonzález,
P.G. Kevrekidis,
I.G. Kevrekidis,
D. Maroudas and
D.J. Frantzeskakis.
Phys. Lett. A
341 (2005) 128134.
Abstract. PDF.
 Precise computations of chemotactic collapse using moving
mesh methods.
C.J.
Budd,
R. CarreteroGonzález and
R.D. Russell.
J. Comput. Phys., 202, 2 (2005) 463487.
Abstract. PDF.
 Families of matterwaves for TwoComponent BoseEinstein
Condensates.
P.G.
Kevrekidis, G.
Theocharis, D.J.
Frantzeskakis, B.A.
Malomed and R.
CarreteroGonzález.
Eur.
Phys. J. D: At. Mol. Opt. Phys., 28,
2, (2004) 181185.
Abstract. PDF.
 Dark soliton dynamics in spatially inhomogeneous media: Application
to BoseEinstein condensates.
G. Theocharis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P.G. Kevrekidis
and
B.A. Malomed.
Math.
Comput. Simulat. 69 (2005) 537552.
Abstract. PDF.
 Vortices in a BoseEinstein
condensate confined by an optical lattice.
P.G.
Kevrekidis, R.
CarreteroGonzález, G.
Theocharis, D.J.
Frantzeskakis and B.A.
Malomed.
J.
Phys. B: At. Mol. Phys. 36
(2003) 34673476.
Abstract. PDF.
 Stability of dark solitons in a BoseEinstein condensate
trapped in an optical lattice.
P.G.
Kevrekidis, R.
CarreteroGonzález, G.
Theocharis, D.J.
Frantzeskakis and B.A.
Malomed.
Phys. Rev. A,
68
035602 (2003).
Abstract. PDF,
[ERRATUM].
 Variational Mesh Adaptation
Methods for Axisymmetrical Problems with Applications to Blowup.
W. Cao,
R. CarreteroGonzález,
W. Huang and
R.D. Russell.
SIAM
Journal on Numerical Analysis 41,1
(2003) 235257.
Abstract. PDF.
 Localized breathing oscillations for BoseEinstein condensates in periodic traps.
R. CarreteroGonzález and
K. Promislow.
Phys.
Rev. A
66, 3,
033610 (2002).
Abstract. PDF.
 Stability of attractive
BoseEinstein condensates in a periodic potential.
J.C. Bronski,
L.D. Carr,
R. CarreteroGonzález,
B. Deconinck,
J.N. Kutz and
K. Promislow.
Phys. Rev. E
64, 5,
056615 (2001).
Abstract. PDF.
 Modelling desert dune
fields based on discrete dynamics.
H. Momiji,
S.R. Bishop,
R. CarreteroGonzález and
A. Warren.
Discrete Dynamics in Nature and Society, 7,
1 (2002)
717.
Abstract. PDF.
 Simulation of the
effect of wind speedup in the formation of transverse dune fields.
H. Momiji,
R. CarreteroGonzález,
S.R. Bishop and
A. Warren.
Earth Surface Processes and Landforms,
25, 8 (2000) 905918.
Abstract. PDF.
 Quasidiagonal approach
to the estimation of Lyapunov spectra for spatiotemporal systems from multivariate
time series.
R. CarreteroGonzález,
S. Řrstavik and
J. Stark.
Phys. Rev. E
62, 5 (2000)
64296439. Abstract. PDF.
 Thermodynamic limit
from small lattices of coupled maps.
R. CarreteroGonzález,
S. Řrstavik,
J. Huke,
D.S. Broomhead and
J. Stark.
Phys. Rev. Lett.
83, 18 (1999) 36333636.
Abstract, PDF.
 Estimation of intensive
quantities in spatiotemporal systems from timeseries.
S. Řrstavik,
R. CarreteroGonzález and
J. Stark.
Physica D
147 (2000) 204220.
Abstract. PDF.
 Scaling and interleaving
of subsystem Lyapunov exponents for spatiotemporal systems.
R. CarreteroGonzález,
S. Řrstavik and
J. Stark.
Chaos
9,
2 (1999)
466482 .
Abstract, PDF.
 Onedimensional dynamics
for travelling fronts in coupled map lattices.
R. CarreteroGonzález,
D.K. Arrowsmith and
F. Vivaldi.
Phys. Rev. E
61, 2 (2000)
13291336.
Abstract, PDF.
 Low dimensional travelling
interfaces in coupled map lattices.
R. CarreteroGonzález.
Int. J. Bifurcation and Chaos
7, 12 (1997)
27452754.
Abstract, PDF.
 ModeLocking in Coupled
Map Lattices.
R. CarreteroGonzález,
D.K. Arrowsmith and
F. Vivaldi.
Physica D
103, 1/4 (1997) 381403.
Abstract, PDF.
 Regular and Chaotic
Behaviour in an Extensible Pendulum.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Eur. J. Phys.
15, 3 (1994)
139148.
Abstract, PDF.
 Evidence of Chaotic
Behaviour in JordanBransDicke Cosmology.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Phys. Lett. A
188, 1 (1994) 4854.
Abstract, PDF.
 Energy localization and transport in twodimensional electrical lattices.
L.Q. English,
F. Palmero,
J. Stormes,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
2013 Int. Symposium on Nonlinear Theory & Its Applications, Santa Fe, New Mexico, USA, September 812, 2013.
Abstract. PDF.
 Optical Manipulation of Matter Waves.
R. CarreteroGonzález,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
B.A. Malomed.
Proc. SPIE Int. Soc. Opt. Eng. 5930 (2005) 59300L.
Abstract. PDF.
 Multifrequency Pattern Generation Using GroupSymmetric Circuits.
J. Neff,
V. In,
B. Meadows,
C. Obra,
A. Palacios, and
R. CarreteroGonzález,
2006 IEEE International Symposium on Circuits and Systems.
Abstract. PDF.
 The Curvature Criterion
and the Dynamics of a Rolling Elastic Cylinder.
M. Arizmendi,
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Proceedings of the "International Symposium on Hamiltonian Systems and Celestial
Mechanics", Cocoyoc, Morelos, México, September 1317, 1994.
Advanced Series in Nonlinear Dynamics, Vol. 8. New Trends for Hamiltonian
Systems and Celestial Mechanics.
E.A.Lacomba & J.Llibre (Eds.). Wolrd Scientific, July (1996) 113. Abstract
 Nonlinear behaviour
in the JBD scalartensor theory.
R. CarreteroGonzález,
P. Chauvet,
H.N. NúńezYépez and
A.L. SalasBrito.
Proceedings of the "International Conference on Aspects of General Relativity
and Mathematical Physics". Mexico City, June 24, 1993.
N.Bretón, R.Capovilla & T.Matos (Eds). CINVESTAV, Mexico City, (1994)
204209. Abstract
 Chaotic behaviour in
JBD cosmology.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Proceedings of the "8th Latin American Symposium on Relativity and Gravitation",
Aguas de Lindoia, Brazil July 2530, 1993.
Gravitation: The Spacetime Structure.
P.S.Letelier & W.A.Rodrigues (Eds.). Wolrd Scientific, July (1994) 457461.
Abstract
 Front propagation
and modelocking in coupled map lattices.
R. CarreteroGonzález.
Ph.D. thesis, Dep. of Mathematical Sciences,
Queen Mary and Westfield College, London, UK, August 1997.
Location at Queen
Mary and Westfield College Library,
Abstract,
PDF,
Table of contents.
 The transition to chaos
on an extensible pendulum.
R. CarreteroGonzález.
B.Sc. Thesis, Facultad de Ciencias, Universidad Nacional Autónoma de
México, México, December 1992. Abstract
Abstracts
[top]
Stability of single and multiple matterwave dark solitons in collisionally
inhomogeneous BoseEinstein condensates.
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Submitted, August 2016.
PDF.
We examine the spectral properties of single and multiple matterwave dark solitons in BoseEinstein
condensates confined in parabolic traps, where the scattering length is periodically modulated.
In addition to the largedensity limit picture previously established for homogeneous nonlinearities,
we explore a perturbative analysis in the vicinity of the linear limit, which provides
very good agreement with the observed spectral modes. Between these two analytically tractable
limits, we use numerical computations to fill in the relevant intermediate regime. We find that the
scattering length modulation can cause a variety of features absent for homogeneous nonlinearities.
Among them, we note the potential oscillatory instability even of the single dark soliton, the potential
absence of instabilities in the immediate vicinity of the linear limit for two dark solitons, and the
existence of an exponential instability associated with the inphase motion of three dark solitons.
[top]
On the spontaneous timereversal symmetry breaking in synchronouslypumped passive Kerr resonators.
J. Rossi,
R. CarreteroGonzález.
P.G. Kevrekidis, and
M. Haragus.
Submitted, May 2016.
PDF.
We study the spontaneous temporal symmetry breaking instability in a
coherentlydriven passive optical Kerr resonator observed experimentally
by Xu and Coen in Opt. Lett. 39, 3492 (2014). We perform a
detailed stability analysis of the LugiatoLefever model for the
optical Kerr resonators and analyze the temporal bifurcation structure
of stationary symmetric and the emerging asymmetric states as a function
of the pump power. For intermediate pump powers a pitchfork loop is
responsible for the destabilization of symmetric states towards stationary
asymmetric ones while at large pump powers we find the emergence of periodic
asymmetric solutions via a Hopf bifurcation. From a theoretical perspective,
we use local bifurcation theory in order to analyze the most unstable
eigenmode of the system. We also explore a nonconservative variational
approximation capturing, among others, the evolution of the solution's
amplitude, width and center of mass. Both methods provide insight towards
the pitchfork bifurcations associated with the symmetry breaking.
[top]
Dark spherical shell solitons in threedimensional BoseEinstein condensates:
Existence, stability, and dynamics.
Wenlong Wang,
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. A 93 (2016) 023630.
PDF.
Movie.
In this work we study
spherical shell dark soliton states in threedimensional
atomic BoseEinstein condensates.
Their symmetry is exploited in order to analyze their existence,
as well as that of topologically charged variants of the structures, and,
importantly, to identify their
linear stability Bogoliubovde Gennes spectrum.
We compare our effective onedimensional spherical and twodimensional2D cylindrical computations
with the full threedimensional numerics. An important conclusion is that such
spherical shell solitons can be stable sufficiently close to the linear limit
of the isotropic condensates considered herein. We have also
identified their instabilities leading to
the emergence of vortex line and vortex ring cages.
In addition, we generalize effective particle pictures of lower dimensional
dark solitons and ring dark solitons
to the spherical shell solitons concerning
their equilibrium radius and effective dynamics around it. In this case too, we
favorably compare, qualitatively and quatitatively, the resulting predictions
such as the shell equilibrium radius
with full numerical solutions in three dimensions.
[top]
Robust Vortex Lines, Vortex Rings and Hopfions in 3D BoseEinstein Condensates.
R.N. Bisset,
Wenlong Wang,
C. Ticknor,
R. CarreteroGonzález,
D.J. Frantzeskakis,
L.A. Collins, and
P.G. Kevrekidis.
Phys. Rev. A 92 (2015) 063611.
PDF.
Movies.
Performing a systematic Bogoliubovde Gennes spectral analysis,
we illustrate that stationary vortex lines, vortex rings and more exotic states, such as hopfions,
are robust in threedimensional atomic BoseEinstein condensates, for large parameter intervals.
Importantly, we find that the hopfion can be stabilized in a simple
parabolic trap, without the need for trap rotation or inhomogeneous interactions.
We supplement our spectral analysis by studying the dynamics of such stationary states;
we find them to be robust against significant perturbations of the initial state.
In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode,
but we also observe the corresponding instability dynamics.
Furthermore, deep in the ThomasFermi regime, we investigate the particlelike behavior of
vortex rings and hopfions.
[top]
Weakly Nonlinear Analysis of Vortex Formation in a Dissipative Variant of the GrossPitaevskii Equation.
J.C. Tzou,
P.G. Kevrekidis,
T. Kolokolnikov, and
R. CarreteroGonzález.
SIAM J. Appl. Dyn. Syst. 15 (2016) 904922.
PDF.
For a dissipative variant of the twodimensional GrossPitaevskii equation with
a parabolic trap under rotation, we study a symmetry breaking process that leads
to the formation of vortices. The first symmetry breaking leads to the formation
of many small vortices distributed uniformly near the ThomasFermi radius. The
instability occurs as a result of a linear instability of a vortexfree steady
state as the rotation is increased above a critical threshold. We focus on the
second subsequent symmetry breaking, which occurs in the weakly nonlinear regime.
At slightly above threshold, we derive a onedimensional amplitude equation that
describes the slow evolution of the envelope of the initial instability. We show
that the mechanism responsible for initiating vortex formation is a modulational
instability of the amplitude equation. We also illustrate the role of dissipation
in the symmetry breaking process. All analyses are confirmed by detailed numerical
compuations.
[top]
NonConservative Variational Approximation for Nonlinear Schrödinger Equations.
J. Rossi,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Submitted, Aug 2015.
PDF.
Recently, Galley [Phys. Rev. Lett. 110, 174301 (2013)]
proposed an initial value problem formulation of Hamilton's principle
applied to nonconservative systems. Here, we explore
this formulation for complex partial differential equations of the
nonlinear Schrödinger (NLS) type, examining the
dynamics of the coherent solitary wave structures of such models by means
of a nonconservative variational approximation (NCVA). We compare the formalism of the NCVA to two other variational techniques used in dissipative systems; namely, the perturbed variational approximation and a generalization of the
socalled Kantorovich method. All three variational techniques produce equivalent equations of motion for the perturbed NLS models studied herein.
We showcase the relevance of the NCVA method by exploring test
case examples within the NLS setting including combinations of linear and
density dependent loss and gain. We also present an example applied to
exciton polariton condensates that intrinsically feature loss and a
spatially dependent gain term.
[top]
Generating and Manipulating Quantized Vortices OnDemand in a BoseEinstein Condensate: a Numerical Study.
B. Gertjerenken,
P.G. Kevrekidis,
R. CarreteroGonzález, and
B.P. Anderson,
Phys. Rev. A 93 (2016) 023604.
PDF.
Movies.
We numerically investigate an experimentally viable method
for generating and manipulating ondemand several vortices in
a highly oblate atomic BoseEinstein condensate (BEC) in order to
initialize complex vortex distributions for studies of vortex dynamics.
The method utilizes moving laser beams
to generate, capture and transport vortices inside and outside the BEC.
We examine in detail this methodology and show a wide parameter range
of applicability for the prototypical twovortex case, and show
case examples of producing and manipulating several vortices for
which there is no net circulation, equal numbers of positive and
negative circulation vortices, and for which there is one net quantum of
circulation. We find that the presence of dissipation can help stabilize
the pinning of the vortices on their respective laser beam pinning sites.
Finally, we illustrate how to utilize laser beams as repositories
that hold large numbers of vortices and how to deposit individual
vortices in a sequential fashion in the repositories in order to construct
superfluid flows about the repository beams with several quanta of circulation.
[top]
Bifurcation and Stability of Single and Multiple Vortex Rings in ThreeDimensional BoseEinstein Condensates.
R.N. Bisset,
Wenlong Wang,
C. Ticknor,
R. CarreteroGonzález,
D.J. Frantzeskakis,
L.A. Collins, and
P.G. Kevrekidis.
Phys. Rev. A 92 (2015) 043601.
PDF.
Selected for Phys. Rev. A's Kaleidoscope.
In the present work, we investigate how single and multivortexring
states can emerge from a planar dark soliton in threedimensional (3D)
BoseEinstein condensates (confined in isotropic or anisotropic traps)
through bifurcations. We characterize such bifurcations
quantitatively using a Galerkintype approach, and find good qualitative and
quantitative agreement with our Bogoliubovde Gennes (BdG) analysis.
We also systematically characterize the BdG spectrum
of the dark solitons, using perturbation theory, and obtain a quantitative match with our 3D
BdG numerical calculations.
We then turn our attention to the emergence of
single and multivortexring states. We systematically capture these
as stationary states of the system and quantify their BdG spectra numerically.
We find that although the vortex ring may be unstable when
bifurcating, its instabilities weaken and may even eventually disappear,
for sufficiently large chemical potentials and suitable trap settings.
For instance, we demonstrate the stability of the vortex ring for an
isotropic trap in the largechemicalpotential regime.
[top]
Solitons Riding on Solitons and the Quantum Newton's Cradle.
Manjun Ma,
R. Navarro, and
R. CarreteroGonzález.
Phys. Rev. E 93 (2016) 022202.
PDF.
The reduced dynamics for dark and bright soliton chains in the onedimensional
nonlinear Schrödinger equation is used to study the behavior of collective
compression waves corresponding to Toda lattice solitons.
We coin the term hypersoliton to describe such solitary waves
riding on a chain of solitons.
It is observed that in the
case of dark soliton chains, the formulated reduction dynamics provides an
accurate an robust evolution of travelling hypersolitons.
As an application to BoseEinstein condensates
trapped in a standard harmonic potential,
we study the case of finite dark soliton chain confined at the
center of the trap.
When the central chain is hit by a dark soliton,
the energy is transferred through the chain as a hypersoliton
that in turn ejects a dark soliton on the other end of the chain that,
as it returns from its excursion up the trap, hits the central
chain repeating the process.
This periodic evolution is an analogue of the classical Newton's cradle.
[top]
Stabilization of ring dark solitons in BoseEinstein condensates.
Wenlong Wang,
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
Tsso J. Kaper, and
Manjun Ma.
Phys. Rev. A 92 (2015) 033611.
PDF.
Earlier work has shown that ring dark solitons in twodimensional
BoseEinstein condensates are generically unstable. In this work,
we propose a way of stabilizing the ring dark soliton via a
radial Gaussian external potential. We investigate the
existence and stability of the ring dark soliton upon variations of
the chemical potential and also of the strength of the radial potential.
Numerical results show that the ring dark soliton can be stabilized in
a suitable interval of external potential strengths and chemical potentials.
We also explore different proposed particle pictures considering the ring as a
moving particle and find, where appropriate, results in very good qualitative and
also reasonable quantitative agreement with the numerical findings.
[top]
Proper Orthogonal Decomposition Methods for the Analysis of RealTime Data:
Exploring Peak Clustering in a Secondhand Smoke Exposure Intervention.
V. Berardi,
R. CarreteroGonzález,
N.E. Klepeis,
A. Palacios,
J. Bellettiere,
S. Hugues,
S. Obayashi, and
M.F. Hovell.
J. Comp. Sci. 11 (2015) 102111.
PDF.
This work explores a method for classifying peaks appearing within a dataintensive
timeseries. We summarize a case study from a clinical trial aimed at reducing
secondhand smoke exposure via the installation of air particle monitors in households.
Proper orthogonal decomposition (POD) in conjunction with a kmeans clustering
algorithm assigns each data peak to one of two clusters. Aversive feedback from
the monitors increased the proportion of shortduration, attenuated peaks from
38.8% to 96.6%. For each cluster, a distribution of parameters from a physicsbased
model of airborne particles is estimated. Peaks generated from these distributions
are correctly identified by POD/clustering with >60% accuracy.
[top]
Optoelectronic Chaos in a Simple Light Activated Feedback Circuit.
K.L. Joiner,
F. Palmero, and
R. CarreteroGonzález.
Int. J. Bifurcation and Chaos 26 (2016) 1650080.
PDF.
The nonlinear dynamics of an optoelectronic negative feedback switching
circuit is studied. The circuit, composed of a bulb, a photoresistor, a
thyristor and a linear resistor, corresponds to a nightlight device whose
light is looped back into its light sensor.
Periodic bifurcations and deterministic chaos are obtained by the feedback
loop created when the thyristor switches on the bulb in the absence of
light being detected by the photoresistor and the bulb light is then
looped back into the nightlight to switch it off.
The experimental signal is analyzed using tools of delayembedding
reconstruction that yield a reconstructed attractor with fractional
dimension and positive Lyapunov exponent
suggesting chaotic behavior for some parameter values.
We construct a simple circuit model reproducing experimental results
that qualitatively matches the different dynamical regimes
of the experimental apparatus.
In particular, we observe an orderchaosorder transition as
the strength of the feedback is varied corresponding to varying
the distance between the nightlight bulb and its photodetector.
A twodimensional parameter diagram of the model reveals that
the orderchaosorder transition is generic for this system.
[top]
Vortex Nucleation in a Dissipative Variant of the Nonlinear Schrödinger Equation under Rotation.
R. CarreteroGonzález,
P.G. Kevrekidis, and
T. Kolokolnikov.
Physica D 317 (2016) 114.
PDF.
Movies.
In the present work, we motivate and explore the dynamics of a dissipative
variant of the nonlinear Schrödinger equation under the impact of
external rotation. As in the well established Hamiltonian case, the rotation
gives rise to the formation of vortices. We show, however, that the most
unstable mode leading to this instability scales with an appropriate
power of the chemical potential μ of the system, increasing
proportionally to μ^{2/3}. The precise form of the relevant formula,
obtained through our asymptotic analysis, provides the most unstable mode as
a function of the atomic density and the trap strength.
We show how these unstable modes typically nucleate a large number
of vortices in the periphery of the atomic cloud.
However, through a pattern selection mechanism, prompted by symmetrybreaking,
only few isolated vortices are pulled in sequentially from the periphery
towards the bulk of the cloud resulting in highly symmetric stable vortex
configurations with far fewer vortices than the original unstable mode.
These results may be of relevance to the
experimentally tractable realm of finite temperature atomic condensates.
[top]
Dynamics of vortex dipoles in anisotropic BoseEinstein condensates.
R.H. Goodman,
P.G. Kevrekidis, and
R. CarreteroGonzález.
SIAM J. Appl. Dyn. Syst. 14 (2015) 699729.
PDF.
We study the motion of a vortex dipole in a BoseEinstein condensate confined
to an anisotropic trap. We focus on a system of ordinary
differential equations describing the vortices' motion, which is in turn a reduced model
of the GrossPitaevskii equation describing the condensate's motion.
Using a sequence of canonical changes of variables, we reduce the
dimension and simplify the equations of motion. We uncover two interesting
regimes. Near a family of periodic orbits known as guiding centers, we find
that the dynamics is essentially that of a pendulum coupled to
a linear oscillator, leading to stochastic reversals in the overall
direction of rotation of the dipole. Near the separatrix orbit in the
isotropic system, we find other families of periodic, quasiperiodic,
and chaotic trajectories. In a neighborhood of the guiding
center orbits, we derive an explicit iterated map that simplifies the
problem further. Numerical calculations are used to illustrate the
phenomena discovered through the analysis. Using the results from the
reduced system we are able to construct complex periodic orbits in
the original, partial differential equation, meanfield model for
BoseEinstein condensates, which corroborates the phenomenology
observed in the reduced dynamical equations.
[top]
Scattering and leapfrogging of vortex rings in a superfluid.
R.M. Caplan,
J.D. Talley,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Fluids 26 (2014) 097101.
PDF.
The dynamics of vortex ring pairs in the homogeneous nonlinear Schr\366dinger equation
is studied. The generation of numerically exact solutions of traveling vortex
rings is described and their translational velocity compared to revised analytic approximations.
The scattering behavior of coaxial vortex rings with opposite charge
undergoing collision is numerically investigated for different scattering angles yielding
a surprisingly simple result for its dependence as a function of the initial vortex
ring parameters.We also study the leapfrogging behavior of coaxial rings with equal
charge and compare it with the dynamics stemming from amodified version of the reduced
equations of motion from a classical fluid model derived using the BiotSavart law.
[top]
Dynamic and Energetic Stabilization of Persistent Currents in BoseEinstein Condensates.
K.J.H. Law,
T.W. Neely,
P.G. Kevrekidis,
B.P. Anderson,
A.S. Bradley, and
R. CarreteroGonzález.
Phys. Rev. A 89 (2014) 053606.
PDF.
We study conditions for which vortices in a highly oblate harmonically trapped
BoseEinstein condensate (BEC) can be stabilized due to pinning by a
bluedetuned Gaussian laser beam, with
particular interest in the potentially destabilizing effects of laser beam positioning
within the BEC. Our approach involves theoretical and numerical exploration of
dynamically and energetically stable pinning of vortices with winding number up
to S, in correspondence with experimental observations.
Stable pinning is quantified theoretically via Bogoliubovde Gennes
excitation spectrum computations and confirmed via direct numerical simulations
for a range of conditions similar to those of experimental observations. The
theoretical and numerical results indicate that the pinned winding number, or
equivalently the winding number of the superfluid current about the laser beam,
decays as the laser beam moves away from the BEC center. Our theoretical analysis
helps explain previous experimental observations, and helps define limits of stable
vortex pinning for future experiments involving vortex manipulation by laser beams.
[top]
A Tale of Two Distributions: From Few To Many Vortices In QuasiTwoDimensional BoseEinstein Condensates.
T. Kolokolnikov,
P.G. Kevrekidis, and
R. CarreteroGonzález.
Proc. R. Soc. A 470 (2014) 20140048.
PDF.
Motivated by the recent successes of particle models in capturing the
precession and interactions of vortex structures in quasitwodimensional
BoseEinstein condensates, we revisit the relevant systems of ordinary
differential equations. We consider the number of vortices N as a
parameter and
explore the prototypical configurations (``ground states'') that arise in the
case of few or many vortices. In the case of few vortices, we modify the
classical result of Havelock [Phil. Mag. 11, 617 (1931)] illustrating
that vortex polygons in the form of a ring are unstable for N ≥ 7.
Additionally, we reconcile this modification with the recent identification of
symmetry breaking bifurcations for the cases of N=2,...,5.
We also briefly discuss the case of a ring of vortices surrounding
a central vortex (socalled N+1 configuration).
We finally examine
the opposite limit of large N and illustrate how a coarsegraining,
continuum approach enables the accurate identification of the radial
distribution of vortices in that limit.
[top]
Exploring Vortex Dynamics in the Presence of Dissipation: Analytical and Numerical Results.
D. Yan,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P.G. Kevrekidis,
N.P. Proukakis, and
D. Spirn.
Phys. Rev. A 89 (2014) 043613.
PDF.
In this paper, we examine the dynamical properties
of vortices in atomic BoseEinstein condensates in the presence of phenomenological dissipation,
used as a basic model for the effect of finite temperatures.
In the context of this socalled dissipative GrossPitaevskii model, we derive analytical results
for the motion of single vortices and, importantly, for vortex dipoles which have become very relevant
experimentally. Our analytical results are shown to compare favorably to the full numerical solution
of the dissipative GrossPitaevskii equation in parameter
regimes of experimental relevance. We also present
results on the stability of vortices and vortex dipoles,
revealing good agreement between numerical and analytical results for
the internal excitation eigenfrequencies, which extends even beyond the regime of validity of this equation for cold atoms.
[top]
Nonlinear PTSymmetric models Bearing Exact Solutions.
H. Xu,
P.G. Kevrekidis,
Q. Zhou,
D.J. Frantzeskakis,
V. Achilleos, and
R. CarreteroGonzález.
Romanian J. Phys. 59 (2014) 185194.
PDF.
We study the nonlinear
Schrödinger
equation with a PTsymmetric potential.
Using a hydrodynamic formulation and
connecting the phase gradient to the field amplitude,
allows for a reduction of the model to a
Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions
existing in the presence of suitable PTsymmetric potentials, and study their stability and dynamics.
We report interesting new features, including oscillatory instabilities of
solitons and (nonlinear) PTsymmetry breaking
transitions, for focusing and defocusing nonlinearities.
[top]
Directed Ratchet Transport in Granular Chains.
V. Berardi,
J. Lydon,
P.G. Kevrekidis,
C. Daraio, and
R. CarreteroGonzález.
Phys. Rev. E 88 (2013) 022912.
PDF.
Directedratchet transport (DRT) in a onedimensional lattice of spherical beads,
which serves as a prototype for granular chains, is investigated. We consider
a system where the trajectory of the central bead is prescribed by a biharmonic
forcing function with broken timereversal symmetry. By comparing the mean
integrated force of beads equidistant from the forcing bead, two distinct types
of directed transport can be observed spatial and temporal DRT.
Based on the value of the frequency of the forcing function relative to the cutoff
frequency, the system can be categorized by the presence and magnitude of each type of
DRT. Furthermore, we investigate and quantify
how varying additional parameters such as the
biharmonic weight affects DRT velocity and magnitude.
Finally, friction is introduced into the system and is found to
significantly inhibit spatial DRT. In fact,
for sufficiently low forcing frequencies, the friction may even
induce a switching of the DRT direction.
[top]
From Nodeless Clouds and Vortices to Gray Ring Solitons and SymmetryBroken States in TwoDimensional Polariton Condensates.
A.S. Rodrigues,
P.G. Kevrekidis,
R. CarreteroGonzález,
J. Cuevas,
D.J. Frantzeskakis, and
F. Palmero,
J. Phys.: Condens. Matter 26 (2014) 155801.
PDF.
Movies.
We consider the existence, stability and dynamics
of the nodeless state and fundamental
nonlinear excitations, such as vortices,
for a quasitwodimensional polariton condensate
in the presence of pumping and nonlinear damping.
We find a series of interesting features that can be directly contrasted
to the case of the typically
energyconserving ultracold alkaliatom
BoseEinstein condensates (BECs). For sizeable parameter ranges,
in line with earlier findings,
the nodeless state becomes unstable
towards the formation of stable nonlinear single or multivortex
excitations. The potential instability of the single vortex is also
examined and is found to possess similar characteristics to those
of the nodeless cloud.
We also report that, contrary to what is known, e.g., for the atomic
BEC case, stable stationary
gray ring solitons (that can be thought of as radial forms of NozakiBekki
holes) can be found for polariton condensates
in suitable parametric regimes. In other regimes, however, these
may also suffer symmetry breaking instabilities.
The dynamical, patternforming implications of the above instabilities
are explored through direct numerical simulations and, in turn, give rise
to waveforms with triangular or quadrupolar symmetry.
[top]
Exploring Rigidly Rotating Vortex Configurations and their Bifurcations in Atomic BoseEinstein Condensate.
A.V. Zampetaki,
R. CarreteroGonzález,
P.G. Kevrekidis,
F.K. Diakonos, and
D.J. Frantzeskakis.
Phys. Rev. E 88 (2013) 042914.
PDF.
In the present work, we consider the problem of a system of few vortices
N≤5 as it emerges from its experimental realization in the
field of atomic BoseEinstein condensates. Starting from the corresponding
equations of motion, we use a twopronged approach in order to reveal the
configuration space of the system's preferred dynamical states. On the one
hand, we use a MonteCarlo method parametrizing the vortex ``particles''
by means of hyperspherical coordinates and identifying the minimal energy
ground states thereof for N=2,...,5 and different vortex
particle angular momenta. We then complement this picture
with a dynamical systems analysis of the possible rigidly rotating states.
The latter reveals all the supercritical and subcritical pitchfork, as well
as saddlecenter bifurcations that arise exposing the full wealth of the
problem even at such low dimensional cases. By corroborating the results
of the two methods, it becomes fairly transparent which branch the MonteCarlo
approach selects for different values of the angular momentum which is used
as a bifurcation parameter.
[top]
PhaseShift Plateaus in the Sagnac Effect for Matter Waves.
M.C. Kandes,
R. CarreteroGonzález, and
M.W.J. Bromley.
Submitted, 2013.
PDF.
The Sagnac effect was first demonstrated experimentally for
light one hundred years ago by French physicist Georges Sagnac
and, in recent years, atoms have begun to exhibit a rotation measurement
sensitivity able to go beyond that of lightbased systems.
We simulate ultracold Sagnac atom interferometers using
quantummechanical matter wavepackets,
e.g. BoseEinstein condensates (BECs),
that counterpropagate within a rotating ringtrap.
We find that the accumulation of the relative phase difference between
wavepackets, i.e. the matter wave Sagnac effect,
is manifested as discrete phase jumps.
We show that the plateaus persist in the presence of
nonlinear atomatom interactions, and in atoms undergoing various rotations,
and thus will occur during matter wavepacket experiments.
We also introduce the simplest possible Sagnac atom interferometry scheme
which relies on wavepacket dispersion around a ringtrap.
[top]
Dynamics of Few Corotating Vortices in BoseEinstein Condensates.
R. Navarro,
R. CarreteroGonzález,
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
M.W. Ray,
E. Altuntaş, and
D.S. Hall.
Phys. Rev. Lett. 110 (2013) 225301.
PDF.
We study the dynamics of small vortex clusters with few (24)
corotating vortices in BoseEinstein condensates
by means of experiments, numerical computations, and theoretical analysis.
All of these approaches corroborate
the counterintuitive presence of a dynamical instability of
symmetric vortex configurations. The instability arises as a
pitchfork bifurcation at sufficiently large values of the
angular momentum that induces the emergence and
stabilization of asymmetric rotating vortex configurations.
The latter are quantified in the theoretical model and
observed in the experiments. The dynamics is explored
both for the integrable twovortex system, where a reduction
of the phase space of the system provides valuable insight, as well as for
the nonintegrable three (or more) vortex
case, which additionally admits the possibility of chaotic trajectories.
[top]
Inelastic Collisions of Solitary Waves in Anisotropic BoseEinstein Condensates:
SlingShot Events and Expanding Collision Bubbles.
C. Becker,
K. Sengstock,
P. Schmelcher,
R. CarreteroGonzález, and
P.G. Kevrekidis.
New J. Phys. 15 (2013) 113028.
PDF.
We study experimentally and theoretically
the dynamics of apparent dark soliton stripes in an
elongated BoseEinstein condensate
referring to a recent experimental setup
for a single repulsive component
[C. Becker et al., Nature Phys. 4, 496 (2008)].
We show that for the trapping strengths corresponding
to our experimental setup, the transverse confinement along one
of the tight directions is not strong enough to arrest the formation
of solitonic vortices or vortex rings.
These solitonic vortices and vortex rings, when integrated along the transverse
direction, appear as dark soliton stripes along the longitudinal direction thereby
hiding their true character. The latter significantly
modifies the interaction dynamics during collision events and
can lead to apparent examples of inelasticity and what may appear
experimentally even as a merger of two dark soliton stripes. We explain this feature by means of the interaction
of two solitonic vortices leading to a sling shot event with one of the solitonic vortices being
ejected at a relatively large speed. Furthermore we observe
expanding collision bubbles which consist of repeated inelastic
collisions of a dark soliton stripe pair with an increasing time interval
between collisions.
[top]
Nonlinear localized modes in twodimensional electrical lattices.
L.Q. English,
F. Palmero,
J. Stormes,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. E 88 (2013) 022912.
PDF.
We report the observation of spontaneous localization of energy in
two spatial dimensions in the context of nonlinear electrical
lattices. Both stationary and traveling selflocalized
modes were generated experimentally and theoretically
in a family of twodimensional square, as well as
honeycomb lattices composed of 6x6 elements.
Specifically, we find regions in driver voltage and frequency
where stationary discrete breathers, also known as intrinsic
localized modes (ILM), exist and are stable due to the interplay
of damping and spatially homogeneous driving. By introducing
additional capacitors into the unit cell, these lattices
can controllably induce traveling discrete breathers.
When more than one such ILMs are experimentally generated in
the lattice, the interplay of nonlinearity, discreteness and
wave interactions generate a complex dynamics wherein the ILMs
attempt to maintain a minimum distance between one another.
Numerical simulations show good agreement with experimental
results, and confirm that these phenomena qualitatively carry
over to larger lattice sizes.
[top]
Solitons and their ghosts in PTsymmetric systems with defocusing
nonlinearities.
V. Achilleos,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity
7, (2014) 342.
PDF.
We examine a prototypical nonlinear Schrödinger model bearing a defocusing
nonlinearity and ParityTime (PT) symmetry.
For such a model, the solutions
can be identified numerically and characterized in the
perturbative limit of small gain/loss. There we find two fundamental
phenomena. First, the dark solitons
that persist in the presence
of the PTsymmetric potential are destabilized via a
symmetry breaking (pitchfork) bifurcation. Second, the ground state and
the dark soliton die handinhand in a saddlecenter bifurcation
(a nonlinear analogue of the PTphase transition)
at a second critical value of the gain/loss parameter. The daughter
states arising from the pitchfork are identified as ``ghost states'',
which are not exact solutions of the original system, yet
they play a critical role in the system's dynamics.
A similar phenomenology is also pairwise identified for higher excited
states, with e.g. the twosoliton structure bearing similar characteristics
to the zerosoliton one, and the threesoliton state having the same
pitchfork destabilization mechanism and saddlecenter collision (in this
case with the twosoliton) as the onedark soliton.
All of the above notions are generalized in twodimensional settings for
vortices, where the topological charge enforces the destabilization of a twovortex
state and the collision of a novortex state with a twovortex one,
of a onevortex state with a threevortex one, and so on.
The dynamical manifestation of the instabilities mentioned above
is examined through direct numerical simulations.
[top]
Symmetrybreaking Effects for Polariton Condensates in DoubleWell Potentials.
A.S. Rodrigues,
P.G. Kevrekidis,
J. Cuevas,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Progress in Optical Science and Photonics, 1 (2013) 509529.
PDF.
We study the existence, stability, and dynamics of symmetric and
antisymmetric states of quasionedimensional polariton condensates in
doublewell potentials, in the presence of nonresonant pumping and nonlinear damping.
Some prototypical features of the system, such as the
bifurcation of asymmetric solutions, are similar to
the Hamiltonian analog of the doublewell system
considered in the realm of atomic condensates. Nevertheless,
there are also some nontrivial differences including, e.g., the
unstable nature of both the parent and the daughter branch emerging
in the relevant pitchfork bifurcation for slightly larger
values of atom numbers. Another interesting feature that does not
appear in the atomic condensate case is that the bifurcation for attractive
interactions is slightly subcritical instead of supercritical.
These conclusions of the bifurcation analysis are corroborated by direct
numerical simulations examining the dynamics of the system in
the unstable regime.
[top]
Characteristics of TwoDimensional Quantum Turbulence in a Compressible Superfluid.
T.W. Neely,
A.S. Bradley,
E.C. Samson,
S.J. Rooney,
E.M. Wright,
K.J.H. Law,
R. CarreteroGonzález,
P.G. Kevrekidis,
M.J. Davis, and
B.P. Anderson.
Phys. Rev. Lett. 111 (2013) 235301.
PDF.
[Supplemental material].
[Movie].
Under suitable forcing a fluid exhibits turbulence, with characteristics
strongly affected by the fluid's confining geometry. Here we study
twodimensional quantum turbulence in a highly oblate BoseEinstein
condensate in an annular trap. As a compressible quantum fluid, this
system affords a rich phenomenology, allowing coupling between vortex
and acoustic energy. Smallscale stirring generates an experimentally
observed disordered vortex distribution that evolves into largescale
flow in the form of a persistent current. Numerical simulation of the
experiment reveals additional characteristics of twodimensional quantum
turbulence: spontaneous clustering of samecirculation vortices, and an
incompressible energy spectrum with k^{5/3} dependence for low
wavenumbers k and k^{3} dependence for high k.
[top]
Dark solitons and vortices in PTsymmetric nonlinear media:
from spontaneous symmetry breaking to nonlinear PT phase transitions.
V. Achilleos,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Phys. Rev. A 86 (2012) 013808.
PDF.
We consider nonlinear analogues of ParityTime
(PT) symmetric linear systems
exhibiting defocusing nonlinearities.
We study the ground state and excited
states (dark solitons and vortices)
of the system and report the following remarkable features.
For relatively weak values of the parameter ε controlling the strength
of the PTsymmetric potential, excited states undergo
(analytically tractable) spontaneous symmetry breaking;
as ε
is further increased, the ground state and first excited state,
as well as branches of higher multisoliton (multivortex) states,
collide in pairs and disappear in bluesky bifurcations, in a way
which is strongly reminiscent of the linear PTphase transition
thus termed the nonlinear PTphase transition.
Past this critical point, initialization of, e.g., the former ground state
leads to spontaneously emerging solitons and vortices.
[top]
Vortices in BoseEinstein Condensates: (Super)fluids with a twist.
P.G. Kevrekidis.
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Dynamical Systems Magazine, October 2011.
WWW.
In this brief exposition, we showcase some recent experimental and theoretical work
in the coldest temperatures in the universe involving topological
defects (vortices) in the newest state of matter: the atomic BoseEinstein condensates.
The remarkable feature that these experiments and associated analysis
illustrate is the existence of a new kind of ``classical mechanics''
for vortices, which revisits the integrability of the
twobody (i.e., twovortex) system and opens up exciting extensions for
Nbody generalizations thereof. Onedimensional and threedimensional
analogues of such dynamics, involving dark solitons and vortex rings,
respectively, are also briefly touched upon.
[top]
A ModulusSquared Dirichlet Boundary Condition for
TimeDependent Complex Partial Differential Equations and
its Application to the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
SIAM J. Sci. Comput., 36 (2014) A1A19.
PDF.
An easy to implement modulussquared Dirichlet (MSD) boundary condition
is formulated for numerical simulations of timedependent complex partial
differential equations in multidimensional settings. The MSD boundary
condition approximates a constant modulussquare value of the solution at the
boundaries. Application of the MSD boundary condition to the nonlinear
Schrödinger equation is shown, and numerical simulations are performed
to demonstrate its usefulness and advantages over other simple boundary conditions.
[top]
A TwoStep HighOrder Compact Scheme for the Laplacian Operator and its Implementation in an
Explicit Method for Integrating the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
J. Comput. Appl. Math. 251 (2013) 3346.
PDF.
We describe and test an easytoimplement twostep highorder compact (2SHOC) scheme for
the Laplacian operator and its implementation into an explicit finite
difference scheme for simulating the nonlinear Schrödinger equation (NLSE).
Our method relies on a compact `doubledifferencing' which is shown to be computationally
equivalent to standard fourthorder noncompact schemes. Through numerical simulations
of the NLSE using fourthorder RungeKutta, we confirm that our scheme shows the desired
fourthorder accuracy. A computation and storage requirement comparison is made between
the 2SHOC scheme and the noncompact equivalent scheme for both the Laplacian operator
alone, as well as when implemented in the NLSE simulations. Stability bounds are also
shown in order to get maximum efficiency out of the method. We conclude that the modest
increase in storage and computation of the 2SHOC schemes are well worth the advantages
of having the schemes compact, and their ease of implementation makes their use very
useful for practical implementations.
[top]
Numerical Stability of Explicit RungeKutta FiniteDifference Schemes for the Nonlinear Schrödinger Equation.
R.M. Caplan and
R. CarreteroGonzález.
App. Num. Math. 71 (2013) 2440.
PDF.
Awarded the 6th most successful IMACS paper published in 2013 in Applied Numerical Mathematics.
Linearized numerical stability bounds for solving the nonlinear timedependent
Schrödinger equation (NLSE) using explicit finitedifferencing are shown.
The bounds are computed for the fourthorder RungeKutta scheme in time and both
secondorder and fourthorder central differencing in space. Results are given
for Dirichlet, modulussquared Dirichlet, Laplacianzero, and periodic boundary
conditions for one, two, and three dimensions. Our approach is to use standard
RungeKutta linear stability theory, treating the nonlinearity of the NLSE as a
constant. The required bounds on the eigenvalues of the scheme matrices are
found analytically when possible, and otherwise estimated using the Gershgorin
circle theorem.
[top]
Dynamics of Vortex Dipoles in Confined BoseEinstein Condensates.
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P. Schmelcher, and
D.S. Hall.
Phys. Lett. A 375 (2011) 30443050.
PDF.
We present a systematic theoretical analysis of the motion of a pair of straight
counterrotating vortex lines within a trapped BoseEinstein condensate. We
introduce the dynamical equations of motion, identify the associated conserved
quantities, and illustrate the integrability of the ensuing dynamics. The system
possesses a stationary equilibrium as a special case in a class of exact solutions
that consist of rotating guidingcenter equilibria about which the vortex lines
execute periodic motion; thus, the generic twovortex motion can be classified
as quasiperiodic. We conclude with an analysis of the linear and nonlinear
stability of these stationary and rotating equilibria.
[top]
Generation of localized modes in an electrical lattice using subharmonic driving.
L.Q. English,
F. Palmero,
P. Candiani,
J. Cuevas,
R. CarreteroGonzález,
P.G. Kevrekidis, and
A.J. Sievers.
Phys. Rev. Lett. 108 (2012) 084101.
PDF.
We show experimentally and numerically that an intrinsic localized mode (ILM)
can be stably produced (and experimentally observed) via subharmonic,
spatially homogeneous driving in the context of a nonlinear electrical
lattice. The precise nonlinear spatial response of the system has been seen to
depend on the relative location in frequency between the driver frequency,
ω_{d}, and the bottom of the linear dispersion curve, ω_{0}.
If ω_{d}/2 lies just below ω_{0}, then a single ILM can be generated
in a 32node lattice, whereas when ω_{d}/2 lies within the dispersion
band, a spatially extended waveform resembling a train of ILMs results. To our
knowledge, and despite its apparently broad relevance,
such an experimental observation of subharmonically
driven ILMs has not been previously reported.
[top]
Multiple darkbright solitons in atomic BoseEinstein condensates.
D. Yan,
J.J. Chang,
C. Hamner,
P.G. Kevrekidis.
P. Engels,
V. Achilleos,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
Phys. Rev. A 84 (2011) 053630.
PDF.
We present experimental results and a systematic theoretical analysis of
darkbright soliton interactions and multipledarkbright soliton complex
es in atomic twocomponent BoseEinstein condensates. We study analytically
the interactions between twodarkbright solitons in a homogeneous con
densate and, then, extend our considerations to the presence of the trap.
An effective equation of motion is derived for the darkbright soliton center
and the existence and stability of stationary twodarkbright soliton states
is illustrated (with the bright components being either in or outofphase).
The equation of motion provides the characteristic oscillation frequencies of
the solitons, in good agreement with the eigenfrequencies of the anomalous
modes of the system.
[top]
Discrete breathers in a nonlinear electric line: Modeling, Computation and Experiment.
F. Palmero,
L.Q. English,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. E 84 (2011) 026605.
PDF.
We study experimentally and numerically the existence and
stability properties of discrete breathers in a periodic
nonlinear electric line.
The electric line is composed of single cell nodes, containing
a varactor diode and an inductor, coupled together in a periodic
ring configuration through inductors and driven uniformly by a
harmonic external voltage source.
A simple model for each cell is proposed by using a
nonlinear form for the varactor characteristics through
the current and capacitance dependence on the
voltage.
For an electrical line composed of 32 elements, we
find the regions, in driver voltage and frequency,
where npeaked breather solutions exist and
characterize their stability. The results are
compared to experimental measurements with good
quantitative agreement.
We also examine the spontaneous formation
of npeaked breathers through modulational instability of
the homogeneous steady state. The competition between
different discrete breathers seeded by the modulational
instability eventually leads to stationary npeaked
solutions whose precise locations is seen to sensitively
depend on the initial conditions.
[top]
GuidingCenter Dynamics of Vortex Dipoles in BoseEinstein Condensates.
S. Middelkamp,
P.J. Torres,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P. Schmelcher,
D.V. Freilich, and
D.S. Hall.
Phys. Rev. A 84 (2011) 011605(R).
PDF.
A quantized vortex dipole is the simplest vortex molecule, comprising two countercirculating vortex lines
in a superfluid. Although vortex dipoles are endemic in twodimensional superfluids,
the precise details of their dynamics have remained largely unexplored. We present here
several striking observations of vortex dipoles in dilutegas BoseEinstein condensates, and develop a
vortexparticle model that generates vortex line trajectories that are in good agreement with the experimental data.
Interestingly, these diverse trajectories exhibit essentially identical quasiperiodic behavior, in which the vortex
lines undergo stable epicyclic orbits.
[top]
Nonlinear Excitations, Stability Inversions and
Dissipative Dynamics in Quasionedimensional Polariton Condensates.
J. Cuevas,
A.S. Rodrigues,
R. CarreteroGonzález,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. B 83 (2011) 245140.
PDF.
We study the existence, stability and dynamics
of the ground state and nonlinear excitations, in the form of
dark solitons, for a quasionedimensional polariton condensate
in the presence of nonresonant pumping and nonlinear damping.
We find a series of remarkable features that can be directly contrasted
to the case of the typically energyconserving ultracold alkaliatom
BoseEinstein condensates. For some sizeable parameter ranges,
the nodeless ("ground") state becomes unstable
towards the formation of stable nonlinear single or multi
darksoliton excitations. It is also observed that for suitable
parametric choices, the instability of single dark solitons
can nucleate multidarksoliton states.
Also, for other parametric regions, stable asymmetric sawtoothlike
solutions exist.
These are shown to emerge through a symmetrybreaking bifurcation
from bubblelike solutions that we also explore.
We also consider the dragging of a defect through the condensate and the
interference of two initially separated condensates, both of which
are capable of nucleating dark multisoliton dynamical states.
[top]
Variational approximations in discrete nonlinear Schrödinger
equations with nextnearestneighbor couplings.
C. Chong,
R. CarreteroGonzález,
B.A. Malomed, and
P.G. Kevrekidis.
Physica D
240 (2011) 12051212.
PDF.
Solitons of a discrete nonlinear Schrödinger equation which
includes the nextnearestneighbor interactions are studied by
means of a variational approximation and
numerical computations. A large family of multihumped solutions,
including those with a nontrivial phase structure which are a
feature particular to the nextnearestneighbor interaction model,
are accurately predicted by the variational approximation.
Bifurcations linking solutions with the trivial
and nontrivial phase structures are also captured remarkably well,
including a prediction of critical parameter values.
[top]
Controlling directed transport of matterwave solitons using the ratchet effect.
M.A. Rietmann,
R. CarreteroGonzález, and
R. Chacon.
Phys. Rev. A 83 (2011) 053617.
PDF.
We demonstrate that directed transport of bright solitons in a
quasionedimensional BoseEinstein condensate can be reliably controlled by
tailoring a weak optical lattice potential, biharmonic in both space and
time, in accordance with the degree of symmetry breaking mechanism.
By considering the
regime where matterwave solitons are narrow compared to the lattice period,
we propose an analytical estimate for the dependence of the
soliton current on the number of atoms and the biharmonic potential
parameters which is in good agreement with numerical experiments.
[top]
Emergence and Stability of Vortex Clusters in BoseEinstein Condensates: a Bifurcation Approach near the Linear Limit.
S. Middelkamp,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
Physica D,
240 (2011) 14491459.
PDF.
We study the existence and stability properties
of clusters of alternating charge vortices
in repulsive BoseEinstein condensates. It is illustrated
that such states emerge from cascades of symmetrybreaking
bifurcations that can be analytically tracked near the linear limit
of the system via weakly nonlinear fewmode expansions. We present the
resulting states that emerge near the first few eigenvalues of
the linear limit, and illustrate how the nature
of the bifurcations can be used to understand their stability.
Rectilinear, polygonal and diagonal vortex
clusters are only some of the obtained states while mixed
states, consisting of dark solitons and vortex clusters, are
identified as well.
We also explore the evolution of unstable
states and their transient dynamics exploring
configurations of nearby bifurcation branches.
[top]
Vortex Interaction Dynamics in Trapped BoseEinstein Condensates.
P.J. Torres,
R. CarreteroGonzález,
S. Middelkamp,
P. Schmelcher, and
P.G. Kevrekidis,
D.J. Frantzeskakis.
Comm. Pure Appl. Ana.
10 (2011) 15891615.
PDF.
Motivated by recent experiments studying the dynamics of
configurations bearing a small number of vortices in
atomic BoseEinstein condensates (BECs), we
illustrate that such systems can be accurately described by
ordinary differential equations (ODEs) incorporating the precession
and interaction dynamics of vortices in harmonic traps.
This dynamics is tackled in detail at the ODE level, both for
the simpler case of equal charge vortices,
(yet also experimentally relevant) case of opposite charge vortices.
In the former case, we identify the dynamics as being chiefly
quasiperiodic (although potentially periodic), while in the
latter, irregular dynamics may ensue when suitable
external drive of the BEC cloud is also considered. Our analytical findings are
corroborated by numerical computations of the reduced ODE system.
[top]
Dynamics of DarkBright Solitons in CigarShaped BoseEinstein Condensates.
S. Middelkamp,
J.J. Chang,
C. Hamner,
R. CarreteroGonzález,
P.G. Kevrekidis,
V. Achilleos,
D.J. Frantzeskakis,
P. Schmelcher, and
P. Engels.
Phys. Lett. A
375 (2011) 642646.
PDF.
Darkbright (DB) oscillation movies:
[ Movie#1, Fig. 3 ]:
Single DB for parameters in Nature Phys. 4, 496 (2008):
N_{D}=92,432 and N_{B}=7,973, (f_{z},f_{y},f_{x})=(85,133,5.9) Hz,
[ Movie#2, Fig. 4.a ]:
Single DB with bright soliton transverse dynamics:
N_{D}=88,181 and N_{B}=1,058, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.d ]:
Two interacting DBs with outofphase (attractive) bright solitons:
N_{D}=5,243 and N_{B}=817, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.e ]:
Two interacting DBs with inphase (repulsive) bright solitons:
N_{D}=5,331 and N_{B}=907, (f_{z},f_{y},f_{x})=(133,133,5.9) Hz.
We explore the stability and dynamics of darkbright solitons in twocomponent
elongated BoseEinstein condensates by developing effective 1D vector equations
as well as solving the corresponding 3D GrossPitaevskii equations. A strong
dependence of the oscillation frequency and of the stability of the darkbright
(DB) soliton on the atom number of its components is found. Spontaneous symmetry
breaking leads to oscillatory dynamics in the transverse degrees of freedom for
a large occupation of the component supporting the dark soliton. Moreover, the
interactions of two DB solitons are investigated with special emphasis on the
importance of their relative phases. Experimental results showcasing darkbright
soliton dynamics and collisions in a BEC consisting of two hyperfine states of
^{87}Rb confined in an elongated optical dipole trap are presented.
[top]
Bifurcations, Stability and Dynamics of Multiple MatterWave Vortex States.
S. Middelkamp,
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález.
P. Schmelcher.
Phys. Rev. A 82 (2010) 013646.
PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 2, issue 8 (2010).
In the present work, we offer a unifying perspective between the dark
soliton stripe, and the vortex multipole (dipole, tripole, aligned quadrupole,
quintopole, etc.) states that emerge in the context of quasitwodimensional
BoseEinstein condensates. In particular, we illustrate that the
multivortex states with the vortices aligned along the (former) dark
soliton stripe sequentially bifurcate from the latter state in a
supercritical pitchfork manner. Each additional
bifurcation adds an extra mode to the dark soliton instability and an
extra vortex to the configuration; also, the
bifurcating states inherit the stability properties of the soliton
prior to the bifurcation.
The critical points of this bifurcation
are computed analytically via a fewmode truncation
of the system, which clearly showcases the symmetrybreaking nature
of the corresponding bifurcation. We complement this
small(er) amplitude, few mode bifurcation picture, with a larger
amplitude, particlebased description of the ensuing vortices.
The latter, enables us to characterize the equilibrium
position of the vortices, as well as
their intrinsic dynamics and anomalous modes,
thus providing a qualitative description of the
nonequilibrium multivortex dynamics.
[top]
Controlling the transverse instability of dark solitons and
nucleation of vortices by a potential barrier.
Manjun Ma,
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis, and
B.A. Malomed.
Phys. Rev. A 82 (2010) 023621.
PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 2, issue 9 (2010).
We study possibilities to suppress the transverse modulational instability
(MI) of darksoliton stripes in twodimensional (2D) BoseEinstein
condensates (BECs) and selfdefocusing bulk optical waveguides by means of
quasi1D structures. Adding an external repulsive barrier potential (which
can be induced in BEC by a laser sheet, or by an embedded plate in optics),
we demonstrate that it is possible to reduce the MI wavenumber band, and
even render the darksoliton stripe completely stable. Using this method, we
demonstrate the control of the number of vortex pairs nucleated by each
spatial period of the modulational perturbation. By means of the
perturbation theory, we predict the number of the nucleated vortices per
unit length. The analytical results are corroborated by the numerical
computation of eigenmodes of small perturbations, as well as by direct
simulations of the underlying GrossPitaevskii/nonlinear
Schrödinger equation.
[top]
Existence, Stability, and Scattering of Bright Vortices in the CubicQuintic Nonlinear Schrödinger Equation.
R.M. Caplan,
R. CarreteroGonzález.
P.G. Kevrekidis, and
B.A. Malomed.
Math. Comput. Simulat. 82 (2012) 11501171.
PDF.
We revisit the topic of the existence and azimuthal modulational
stability of solitary vortices (alias vortex rings) in the
twodimensional (2D) cubicquintic nonlinear Schrödinger
equation. We develop a semianalytical approach, assuming that the
vortex ring is relatively narrow, and approximately splitting the
full 2D equation into radial and azimuthal 1D equations. A
variational approach is elaborated to predict the radial shape of
the vortex soliton, using the radial equation. Previously known
existence bounds for the solitary vortices are recovered by means of
this approach. The azimuthal equation is used to analyze the
modulational instability of the vortex ring against the breakup. The
semianalytical predictions, in particular, that for the critical
intrinsic frequency of the vortex soliton at the instability border,
are compared to systematic direct 2D simulations. We also compare
our findings to those reported in earlier works, which featured some
discrepancies. Also, detailed computational results are presented for
collisions and scattering between stable vortices with different topological
charges. In particular, borders between elastic and destructive
collisions are identified.
[top]
Stability and dynamics of matterwave vortices in the
presence of collisional inhomogeneities and dissipative perturbations.
S. Middelkamp,
P.G. Kevrekidis, and
D.J. Frantzeskakis,
R. CarreteroGonzález, and
P. Schmelcher.
J. Phys. B
43 (2010) 155303.
PDF.
In this work, the spectral properties of a singlycharged vortex in a
BoseEinstein condensate confined in a highly anisotropic (diskshaped)
harmonic trap are investigated. Special emphasis is given on the analysis
of the socalled anomalous (negative energy) mode of the Bogoliubov
spectrum. We use analytical and numerical techniques to illustrate
the connection of the anomalous mode to the precession dynamics of
the vortex in the trap. Effects due to inhomogeneous interatomic
interactions and dissipative perturbations motivated
by finite temperature considerations are explored. We find that
both of these effects may give rise to oscillatory instabilities of the
vortex, which are suitably diagnosed through the perturbationinduced
evolution of the anomalous mode, and being monitored by direct
numerical simulations.
[top]
Manipulation of Vortices by Localized Impurities in BoseEinstein Condensates.
M.C. Davis,
R. CarreteroGonzález,
Z. Shi,
K.J.H. Law,
P.G. Kevrekidis, and
B.P. Anderson.
Phys. Rev. A 80 (2009) 023604.
PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 1, issue 3 (2009).
We consider the manipulation of BoseEinstein condensate
vortices by optical potentials generated by focused laser beams.
It is shown that for
appropriate choices of the laser strength and width it is
possible to successfully transport vortices to various
positions inside the trap confining the condensate atoms.
Furthermore,
the full bifurcation structure of possible stationary singlecharge
vortex solutions in a harmonic potential with this type of impurity
is elucidated.
The case when a moving vortex is
captured by a stationary laser beam is also studied,
as well as the possibility of dragging the vortex
by means of periodic optical lattices.
[top]
Phase Separation and Dynamics of Twocomponent BoseEinstein Condensates.
R. Navarro,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. A 80 (2009) 023613.
PDF.
Selected for the Virtual Journal of Atomic Quantum Fluids volume 1, issue 3 (2009).
We study the interactions between two atomic species in a binary
BoseEinstein condensate to revisit the conditions for miscibility,
oscillatory dynamics between the species, steady state solutions and their
stability.
By employing a variational approach for a quasi
onedimensional, twoatomic species, condensate we
obtain equations of motion for the
parameters of each species: amplitude, width, position and phase.
A further simplification leads to a reduction of the dynamics
into a simple classical
Newtonian system where components oscillate in an effective
potential with a frequency that
depends on the harmonic trap strength and the interspecies
coupling parameter.
We develop
explicit conditions for miscibility that can be interpreted as a
phase diagram that depends on the harmonic trap's strength and
the interspecies species coupling parameter.
We numerically illustrate the bifurcation scenario whereby
nontopological, phaseseparated states of increasing complexity
emerge out of a symmetric state, as the interspecies coupling
is increased. The symmetrybreaking
dynamical evolution of some of these states is numerically monitored
and the associated asymmetric states are also explored.
[top]
Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation.
R.M. Caplan,
Q.E. Hoq,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Optics. Comm. 282 (2009) 13991405.
PDF.
We study the azimuthal modulational instability of vortices with different
topological charges, in the focusing twodimensional nonlinear Schrödinger
(NLS) equation. The method of studying the stability relies on freezing the
radial direction in the Lagrangian functional of the NLS in order to form
a quasionedimensional azimuthal equation of motion, and then applying a
stability analysis in Fourier space of the azimuthal modes. We formulate
predictions of growth rates of individual modes and find that vortices are
unstable below a critical azimuthal wave number. Steady state vortex
solutions are found by first using a variational approach to obtain an
asymptotic analytical ansatz, and then using it as an initial condition
to a numerical optimization routine. The stability
analysis predictions are corroborated by direct numerical simulations of
the NLS. We briefly show how to extend the method to encompass nonlocal
nonlinearities that tend to stabilize solutions.
[top]
Spinor BoseEinstein condensate past an obstacle.
S. S. Rodrigues,
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P. Schmelcher,
T.J. Alexander, and.
Yu.S. Kivshar.
Phys. Rev. A 79 (2009) 043603.
PDF.
In this work, we investigate the flow of a spinor (F=1) BoseEinstein condensate
in the presence of an obstacle. We consider both cases of ferromagnetic and polar
spindependent interactions and
find that the system possesses two speeds of sound that are
identified analytically. Numerical simulations illustrate the
nucleation of macroscopic nonlinear structures, such as dark solitons and
vortexantivortex pairs, as well as vortex rings in one and higherdimensional
settings respectively, when
a localized defect (e.g., a bluedetuned laser beam) is dragged through the spinor
condensate at a speed larger than the second critical one.
[top]
Dissipative Solitary Waves in Periodic Granular Crystals.
R. CarreteroGonzález,
D. Khatri,
M.A. Porter,
P.G. Kevrekidis, and
C. Daraio.
Phys. Rev. Lett. 102 (2009) 024102.
PDF.
We provide a quantitative characterization of dissipative effects in
onedimensional granular crystals. We use the propagation
of highly nonlinear solitary waves as a diagnostic tool and develop
optimization schemes that allow one to compute the relevant exponents
and prefactors of the dissipative terms in the equations of motion.
We thereby propose a quantitativelyaccurate extension of the Hertzian
model that encompasses
dissipative effects via a discrete Laplacian of the velocities. Experiments
and computations with steel, brass, and polytetrafluoroethylene reveal
a common dissipation exponent
with a materialdependent prefactor.
[top]
Controlling chaos of a BoseEinstein condensate loaded into a moving
optical Fouriersynthesized lattice.
R. Chacon,
D. Bote, and
R. CarreteroGonzález.
Phys. Rev. E 78 (2008) 036215.
PDF.
We study the chaotic properties of steady state traveling wave solutions
of the particle number density of a
BoseEinstein condensate with an attractive interatomic interaction loaded
into a traveling optical lattice of variable shape. We demonstrate
theoretically and numerically that chaotic traveling steady states can be
reliably suppressed by small changes of the traveling optical
lattice shape while keeping the remaining parameters constant. We find that
the regularization route as the optical lattice shape is continuously varied is
fairly rich, including crisis phenomena and period doubling
bifurcations. The conditions for a possible
experimental realization of the control method are discussed.
[top]
A Map Approach to Stationary Solutions of the
Discrete Nonlinear Schrödinger Equation.
R. CarreteroGonzález.
Book chapter for: Discrete Nonlinear Schrödinger Equation:
Mathematical Analysis, Numerical Computations and Physical Perspectives,
P.G. Kevrekidis (Ed),
Springer Tracts in Modern Physics, Vol. 232, 2009.
PDF.
In this chapter we discuss the wellestablished
map approach for obtaining
stationary solutions to the onedimensional (1D) discrete
nonlinear Schrödinger (DNLS) equation.
The method relies on casting the ensuing stationary
problem in the form of a recurrence relationship that can in turn be cast
into a twodimensional (2D) map
Within this description, any orbit for this 2D map
will correspond to a steady state solution of the original
DNLS equation.
The map approach is extremely useful in finding localized
solutions such as bright and dark solitons. As we will
see in what follows, this method allows for a global
understanding of the types of solutions that are present
in the system and their respective bifurcations.
[top]
Structure and stability of twodimensional BoseEinstein condensates
under both harmonic and lattice confinement.
K.J.H. Law,
P.G. Kevrekidis,
B.P. Anderson,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
J. Phys. B,
41 (2008) 195303.
PDF.
In this work, we study pancakeshaped BoseEinstein
condensates confined by both a cylindrically symmetric harmonic
potential and an optical lattice with equal periodicity
in two orthogonal directions. We first identify
the spectrum of the underlying twodimensional linear
problem through multiplescale techniques. Then, we
use the results obtained in the linear limit
as a starting point for a nonlinear
existence and stability analysis of the lowest energy states,
emanating from the linear ones, in the nonlinear problem. Twoparameter continuations
of these states are performed for increasing nonlinearity and optical
lattice strengths, and their instabilities
and temporal evolution are investigated.
It is found that the ground state as well as one of the excited states
are either stable or weakly unstable for both
attractive and repulsive interatomic interactions.
[top]
Surface Solitons in Three Dimensions.
Q.E. Hoq,
R. CarreteroGonzález,
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis,
Yu.V. Bludov, and
V.V. Konotop.
Phys. Rev. E 78 (2008) 036605.
PDF.
We study localized modes on the surface of a threedimensional dynamical
lattice. The stability of these structures on the surface is investigated
and compared to that in the bulk of the lattice. Typically, the surface
makes the stability region larger, an extreme example of that being the
threesite "horseshoe"shaped structure, which is always unstable in the
bulk, while at the surface it is stable near the anticontinuum limit. We
also examine effects of the surface on lattice vortices. For the vortex
placed parallel to the surface this increased stability region feature is
also observed, while the vortex cannot exist in a state normal to the
surface. More sophisticated localized dynamical structures, such as
fivesite horseshoes and pyramids, are also considered.
[top]
Multistable Solitons in HigherDimensional CubicQuintic Nonlinear Schrödinger Lattices.
C. Chong,
R. CarreteroGonzález,
B.A. Malomed, and
P.G. Kevrekidis.
Physica D,
238 (2009) 126136.
PDF.
We study the existence, stability, and mobility of fundamental
discrete solitons in two and threedimensional nonlinear
Schrödinger lattices with a combination of cubic selffocusing
and quintic selfdefocusing onsite nonlinearities. Several species
of stationary solutions are constructed, and bifurcations linking
their families are investigated using parameter continuation starting
from the anticontinuum limit, and also with the help of a
variational approximation. In particular, a new species of hybrid
solitons, intermediate between the site and bondcentered types
of the localized states, is found in 2D and 3D lattices, while its
counterpart in the 1D model does not exist. We also describe the
mobility of multidimensional discrete solitons that
can be set in motion by lending them kinetic energy exceeding the
appropriately crafted PeierlsNabarro barrier; however,
they eventually come to a halt,
due to radiation loss.
[top]
Nonlinear dynamics of Bosecondensed gases by means of a
qGaussian variational approach.
A.I. Nicolin and
R. CarreteroGonzález.
Physica A 387 (2008) 6032.
Abstract. PDF.
PDF.
We propose a versatile variational method to investigate the
spatiotemporal dynamics of onedimensional magneticallytrapped
Bosecondensed gases. To this end we employ a qGaussian
trial wavefunction that interpolates between the low and the
highdensity limit of the ground state of a Bosecondensed gas.
Our main result consists of reducing the GrossPitaevskii
equation, a nonlinear partial differential equation describing the
T=0 dynamics of the condensate, to a set of only three
equations: two coupled nonlinear ordinary differential
equations describing the phase and the curvature of the
wavefunction and a separate algebraic equation yielding
the generalized width. Our equations recover those of the usual
Gaussian variational approach (in the lowdensity regime), and the
hydrodynamic equations that describe the highdensity regime.
Finally, we show a detailed comparison between the numerical
results of our equations and those of the original
GrossPitaevskii equation.
[top]
Solitons in onedimensional nonlinear Schrödinger lattices with a local inhomogeneity.
F. Palmero,
R. CarreteroGonzález,
J. Cuevas,
P.G. Kevrekidis, and
W. Królikowski.
Phys. Rev. E 77 (2008) 036614.
PDF.
In this paper we analyze the existence, stability, dynamical
formation and mobility properties of
localized solutions in a onedimensional system described by the
discrete nonlinear Schrödinger equation with a linear point defect.
We consider both attractive and repulsive defects in a focusing
lattice. Among our main findings are: a) the destabilization of the
onsite mode centered at the defect in the repulsive case; b) the
disappearance of localized modes in the vicinity of the defect due
to saddlenode bifurcations for sufficiently strong defects of either
type; c) the decrease of the amplitude formation threshold for attractive
and its increase for repulsive defects; and d) the detailed elucidation
as a function of initial speed and defect strength of the different
regimes (trapping, trapping and reflection, pure reflection and
pure transmission) of interaction of a moving localized mode with the
defect.
[top]
Dynamics of Vortex Formation in Merging BoseEinstein Condensate Fragments.
R. CarreteroGonzález,
B.P. Anderson,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
C.N. Weiler.
Phys. Rev. A 77 (2008) 033625.
PDF.
We study the formation of vortices in a BoseEinstein
condensate that has been prepared into separated fragments
that are allowed to collide. We focus on the experimental
set up of Scherer et al., Phys. Rev. Lett. 98, 110402 (2007),
where the condensate is separated into three fragments by a laser
sheet. We perform a detailed numerical study in two dimensions of
the effects of the relative phases of the different fragments
and the ramping down of the laser sheet on the vortex formation.
We find that the longer the ramping time, the smaller the number
of ensuing vortices is. We also observe that the relative
ratio of the phases between the fragments (for sufficiently
long ramping times) leaves a clear imprint on the resulting
configuration; near the center of the cloud, we obtain a single
vortex only if the relative phases are in a suitable region of
the corresponding plane.
Finally, we emulate the full threedimensional system and
study the formation of vortex lines and vortex rings due to
the merger of the condensate fragments; our results illustrate
how the relevant vorticity is manifested for appropriate phase
differences, as well as how it may be masked by the planar projections
observed experimentally.
[top]
Resonant energy transfer in BoseEinstein condensates.
A.I. Nicolin,
M.H. Jensen,
J.W. Thomsen, and
R. CarreteroGonzález.
Physica D,
237 (2008) 24762481.
PDF.
We consider the dynamics of a dilute, magneticallytrapped onedimensional
BoseEinstein condensate whose scattering length is periodically
modulated with a frequency that linearly increases in time.
We show that the response frequency
of the condensate locks to its eigenfrequency for
appropriate ranges of the parameters.
The locking sets in at resonance, i.e.,
when the effective frequency of driving field is equal to the eigenfrequency,
and is accompanied by a sudden increase of the oscillations amplitude
due to resonant energy transfer. We show that the dynamics of the
condensate is given, to leading order, by
a driven harmonic oscillator on the
timedependent part of the width of the condensate.
This equation captures accurately both the locking and the resonant
energy transfer as it is evidenced by comparison with direct numerical
simulations of original GrossPitaevskii equation.
[top]
Nonlinear Waves in BoseEinstein Condensates:
Physical Relevance and Mathematical Techniques.
R. CarreteroGonzález,
D.J. Frantzeskakis, and
P.G. Kevrekidis.
Nonlinearity 21 (2008) R139R202.
PDF.
The aim of the present review is to introduce the reader to some of
the physical notions and of the mathematical methods that are relevant to
the study of nonlinear waves
in BoseEinstein Condensates (BECs).
Upon introducing the general framework, we discuss the prototypical
models that are relevant to this setting for different dimensions
and different potentials confining the atoms. We analyze some of
the model properties and explore their characteristic wave solutions
(plane wave solutions, bright, dark, gap solitons, as well as vortices).
We then offer a collection of mathematical methods that can be used
to understand the existence, stability and dynamics of nonlinear waves
in such BECs, either directly or
starting from different types of limits (e.g., the linear
or the nonlinear limit, or the discrete limit of the corresponding equation).
Finally, we consider some special topics involving more recent developments,
and experimental setups in which there is still considerable
need for developing mathematical as well as computational tools.
[top]
Radially Symmetric Nonlinear States of Harmonically Trapped BoseEinstein Condensates.
G. Herring,
L.D. Carr,
R. CarreteroGonzález,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. A 77 (2008) 023625.
PDF.
Starting from the spectrum of the radially symmetric quantum
harmonic oscillator in two dimensions, we create a large set of nonlinear
solutions. The relevant three principal branches, with n_{r}=0,1 and 2
radial nodes respectively, are systematically
continued as a function of the chemical potential and their linear
stability is analyzed in detail, in the
absence as well as in the presence of topological charge m, i.e., vorticity.
It is found that for repulsive interatomic interactions only the ground
state is linearly stable throughout the parameter range examined.
Furthermore, this is true for topological charges m=0 or m=1; solutions
with higher topological charge can be unstable even in that case.
All higher excited states are found to be unstable in a wide parametric
regime. However, for the focusing/attractive case the ground
state with n_{r}=0 and m=0 can only be
stable for a sufficiently low number of atoms.
Once again, excited states are found to be generically unstable.
For unstable profiles,
the dynamical evolution of the corresponding branches is also followed
to monitor the temporal development of the instability.
[top]
Faraday waves in BoseEinstein condensates.
A.I. Nicolin,
R. CarreteroGonzález, and
P.G. Kevrekidis.
Phys. Rev. A 76 (2007) 063609.
PDF.
Motivated by recent experiments on Faraday waves in BoseEinstein
condensates we investigate both analytically and numerically the
dynamics of cigarshaped Bosecondensed gases subject to periodic
modulation of the strength of the transverse confinement. We offer
a fully analytical explanation of the observed parametric
resonance, based on a Mathieutype analysis of the nonpolynomial
Schrödinger equation. The theoretical prediction for the
pattern periodicity versus the driving frequency is directly
compared with the experimental data, yielding good qualitative and
quantitative agreement between the two. These results are
corroborated by direct numerical simulations of both the
onedimensional nonpolynomial Schrödinger equation and of the
fully threedimensional GrossPitaevskii equation.
[top]
Extended Nonlinear Waves in Multidimensional Dynamical Lattices.
Q.E. Hoq,
J. Gagnon,
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Math. Comput. Simulat.,
80 (2009) 721731.
PDF.
We explore spatially extended dynamical states in
the discrete nonlinear Schrödinger lattice in two and three
dimensions, starting from the anticontinuum limit. We first
consider the "core" of the relevant states (either a
twodimensional "tile" or a threedimensional "stone"), and
examine its stability analytically. The predictions are
corroborated by numerical results. When the core is stable, we
propose a method allowing
the extension of the structure to as many
sites as may be desired.
In this way, various patterns of excited
sites can be formed. The stability of the full extended nonlinear
structures is studied numerically, which yields instability
thresholds for such structures, which are attained with the
increase of the lattice coupling constant.
Finally, in cases of instability, direct numerical
simulations are used to elucidate the evolution of the pattern; it is
found that, typically, the unstable extended nonlinear
pattern breaks up in an oscillatory way, leading to "lattice turbulence".
[top]
Polarized States and Domain Walls in Spinor BoseEinstein Condensates.
H.E. Nistazakis,
D.J. Frantzeskakis.
P.G. Kevrekidis,
B.A. Malomed, and
R. CarreteroGonzález, and
A.R. Bishop.
Phys. Rev. A 76 (2007) 063603.
PDF.
We study spinpolarized states and their stability in antiferromagnetic
states of spinor (F=1) quasionedimensional BoseEinstein condensates.
Using analytical approximations and numerical methods, we find various types
of polarized states, including: patterns of the ThomasFermi type;
structures with a pulseshape in one component inducing a hole in
the other components; states with holes in all three components; and domain
walls. A Bogoliubovde Gennes analysis reveals that families of these
states contain intervals of a weak oscillatory instability, except for the domain
walls, which are always stable. The development of the instabilities is
examined by means of direct numerical simulations.
[top]
BrightDark Soliton Complexes in Spinor BoseEinstein Condensates.
H.E. Nistazakis,
D.J. Frantzeskakis.
P.G. Kevrekidis,
B.A. Malomed, and
R. CarreteroGonzález.
Phys. Rev. A 77 (2008) 033612.
PDF.
We present novel solutions for brightdark vector solitons in
quasionedimensional spinor (F=1) BoseEinstein condensates. Using a
multiscale expansion technique, we reduce the corresponding system of three
coupled GrossPitaevskii equations (GPEs) to a completely integrable
YajimaOikawa system. In this way, we obtain approximate solutions for
smallamplitude vector solitons of darkdarkbright and brightbrightdark
types, in terms of the m_{F}=+1,1,0 spinor components, respectively. By
means of numerical simulations of the full GPE system, we demonstrate that
these states feature solitary wave properties, i.e., they propagate
undistorted and undergo quasielastic collisions. It is also shown that, in
the presence of a parabolic trap of strength Ω, the bright
component(s) are guided by the dark one(s), so that the vector soliton as a
whole performs harmonic oscillations of frequency Ω/√2.
[top]
Symmetry breaking in linearly coupled dynamical lattices.
G. Herring,
P.G. Kevrekidis.
B.A. Malomed,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. E 76 (2007) 066606.
PDF.
We examine one and twodimensional (1D and 2D) models of linearly
coupled lattices of the discretenonlinearSchrödinger type.
Analyzing ground states of the systems with equal powers in the
two components, we find a symmetrybreaking phenomenon beyond a
critical value of the squared L^{2}norm. Asymmetric
states, with unequal powers in their components, emerge through a
subcritical pitchfork bifurcation, which, in the limit of a very
weakly coupled lattice, takes a supercritical form. We identify
the stability of various solution branches. Dynamical
manifestations of the symmetry breaking are studied by simulating
the evolution of the unstable branches. The results present the
first ever example of the spontaneous symmetry breaking in 2D
lattice solitons (which has no counterpart in the continuum limit,
because of the collapse instability in that limit).
[top]
NonEquilibrium Dynamics and Superfluid Ring Excitations in Binary BoseEinstein Condensates.
K.M. Mertes,
J. Merrill,
R. CarreteroGonzález,
D.J. Frantzeskakis,
P.G. Kevrekidis, and
D.S. Hall.
Phys. Rev. Lett. 99 (2007) 190402.
PDF.
Movies:
[ cuts @ 60% and 30% for 1> and 2> ],
[ cuts @ 50% and 55% for 1> and 2> ]
We revisit a classic study [D. S. Hall et al., Phys. Rev. Lett. 81, 1539 (1998)]
of interpenetrating BoseEinstein condensates in the hyperfine states
F=1,m_{f}=1>=1> and
F=2,m_{f}=+1>=2>
in ^{87}Rb
and observe striking new nonequilibrium component separation dynamics in the form of
oscillating ringlike structures. The process of component separation is not
significantly damped, a finding that also contrasts sharply with earlier experimental
work, allowing a clean first look at a collective excitation of a binary superfluid.
We further demonstrate extraordinary quantitative agreement between theoretical and
experimental results using a multicomponent meanfield model with key additional
features: the inclusion of atomic losses and the careful characterization of trap
potentials (at the level of a fraction of a percent).
[top]
Čerenkovlike radiation in a binary superfluid flow past an obstacle.
H. Susanto,
P.G. Kevrekidis.
R. CarreteroGonzález,
B.A. Malomed,
D.J. Frantzeskakis, and
A.R. Bishop.
Phys. Rev. A 75 (2007) 055601.
PDF.
We consider the dynamics of two coupled miscible BoseEinstein
condensates, when an obstacle is dragged through them. The existence of
two different speeds of sound provides the possibility for
three dynamical regimes: when both components are subcritical, we do not observe
nucleation of coherent structures; when both components are supercritical
they both form dark solitons in one dimension (1D)
and vortices or rotating vortex dipoles in
two dimensions (2D); in the intermediate regime, we observe
the nucleation of a structure in the form of a darkantidark soliton in 1D;
the 2D analog of such a structure, a vortexlump, is also observed.
[top]
Vortex Structures Formed by the Interference of Sliced Condensates.
R. CarreteroGonzález,
N. Whitaker,
P.G. Kevrekidis, and
D.J. Frantzeskakis.
Phys. Rev. A,
77 (2008) 023605.
PDF.
We study the formation of vortices, vortex necklaces and vortex ring
structures as a result of the interference of higherdimensional BoseEinstein condensates (BECs).
This study is motivated by earlier theoretical results pertaining to
the formation of dark solitons by interfering quasi onedimensional BECs,
as well as recent experiments demonstrating the formation of vortices by
interfering higherdimensional BECs. Here, we
demonstrate the genericity of the relevant scenario, but also
highlight a number of additional
possibilities emerging in higherdimensional settings.
A relevant example is, e.g.,
the formation of a "cage" of vortex rings surrounding
the threedimensional bulk of the condensed atoms.
The effects of the relative phases of the different BEC fragments and
the role of damping due to coupling with the thermal cloud
are also discussed. Our predictions should be
immediately tractable in currently existing experimental BEC setups.
[top]
Mode locking of a driven BoseEinstein condensate.
A.I. Nicolin,
M.H. Jensen, and
R. CarreteroGonzález.
Phys. Rev. E
75 (2007) 036208.
PDF.
We consider the dynamics of a driven BoseEinstein condensate with
positive scattering length. Employing an accustomed variational
treatment we show that when the scattering length is
timemodulated as
a(1+ε sin (ω(t)t)), where
ω(t) increases linearly in time,
i.e., ω(t)=γ t, the response
frequency of the condensate locks to the eigenfrequency for small
values of ε. A simple analytical model is presented which
explains this phenomenon by mapping it to an autoresonance,
i.e., close to resonance
the reduced equations describing the collective behavior of the
condensate are equivalent to those of a virtual particle trapped
in a finitedepth energyminimum of an effective potential.
[top]
Rotating matter waves in BoseEinstein condensates.
T. Kapitula,
P.G. Kevrekidis, and
R. CarreteroGonzález.
Physica D,
233 (2007) 112137.
PDF.
In this paper we consider analytically and numerically the dynamics
of waves in twodimensional, magnetically trapped BoseEinstein
condensates in the weak interaction limit. In particular, we
consider the existence and stability of azimuthally modulated
structures such as rings, multipoles, soliton necklaces, and vortex
necklaces. We show how such structures can be constructed from the
linear limit through LyapunovSchmidt techniques and continued to
the weakly nonlinear regime. Subsequently, we examine their
stability, and find that among the above solutions the only one
which is always stable is the vortex necklace. The analysis is given
for both attractive and repulsive interactions among the condensate
atoms. Finally, the analysis is corroborated by numerical
bifurcation results, as well as by numerical evolution results that
showcase the manifestation of the relevant instabilities.
[top]
Discrete surface solitons in two dimensions.
H. Susanto,
P.G. Kevrekidis,
B.A. Malomed,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Rev. E 75 (2007) 056605.
PDF.
We investigate fundamental localized modes in 2D lattices with an edge (surface).
Interaction with the edge expands the stability area for
ordinary solitons, and induces a difference between perpendicular and
parallel dipoles; on the contrary, lattice vortices cannot
exist too close to the border.
Furthermore, we show analytically and numerically
that the edge stabilizes a novel wave species,
which is entirely unstable in the uniform lattice,
namely, a "horseshoe" soliton, consisting of 3 sites.
Unstable horseshoes transform themselves into a pair
of ordinary solitons.
[top]
Mobility of Discrete Solitons in Quadratic Nonlinear Media.
H. Susanto,
P.G. Kevrekidis,
R. CarreteroGonzález,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. Lett. 99 (2007) 214103.
PDF.
We study the mobility of solitons in lattices with quadratic (χ^{(2)},
alias secondharmonicgenerating) nonlinearity. Using the notion of the
PeierlsNabarro potential and systematic numerical simulations, we
demonstrate that, in contrast with their cubic (χ^{(3)}) counterparts,
the discrete quadratic solitons are mobile not only in the onedimensional
(1D) setting, but also in two dimensions (2D), in any direction. We
identify parametric regions where an initial kick applied to a soliton leads
to three possible outcomes, namely, staying put, persistent motion, or
destruction. On the 2D lattice, the solitons survive the largest kick and
attain the largest speed along the diagonal direction.
[top]
Skyrmionlike states in two and threedimensional dynamical lattices.
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis,
B.A. Malomed, and
F.K. Diakonos.
Phys. Rev. E
75 (2007) 026603.
PDF.
We construct, in discrete twocomponent systems with cubic nonlinearity,
stable states emulating Skyrmions of the classical field
theory. In the 2D case, an analog of the babySkyrmion is
built on the square lattice as a discrete vortex soliton of a
complex field [whose vorticity plays the role of the Skyrmion's
winding number (WN)], coupled to a radial "bubble" in a real
lattice field. The most compact quasiSkyrmion on the cubic
lattice is a toroidal structure, composed of a nearly planar
complexfield discrete vortex and a 3D realfield bubble; unlike
its continuum counterpart which must have WN=2, this
stable discrete state exists with WN=1. Analogs of
Skyrmions in the 1D lattice are also constructed. Stability
regions for all these states are found in an analytical
approximation and verified numerically. The dynamics of unstable
discrete Skyrmions (which leads to onset of lattice turbulence),
and their stabilization by external potentials are explored too.
[top]
Multipolemode solitons in Bessel optical lattices.
Y.V. Kartashov,
R. CarreteroGonzález,
B.A. Malomed,
V.A. Vysloukh, and
Ll. Torner.
Optics Express
13, 26 (2006) 1070310710.
PDF.
We study basic properties of multipolemode solitons supported by the
axially symmetric Bessel lattices in a medium with defocusing cubic
nonlinearity. The spatially localized solitons can be found in
different rings of the lattice. They become stable when the propagation
constant exceeds a critical value, provided that optical lattice is
deep enough. In a highpower limit, the multipolemode solitons feature
a multiring structure.
[top]
Soliton trains and vortex streets as a form of Cerenkov radiation in
trapped BoseEinstein condensates.
R. CarreteroGonzález,
P.G. Kevrekidis,
D.J. Frantzeskakis.
B.A. Malomed,
S. Nandi, and
A.R. Bishop.
Math. Comput. Simulat.,
74 (2007) 361369.
PDF.
We numerically study the nucleation of gray solitons
and vortexantivortex pairs created by a moving
impurity in, respectively, 1D and 2D
BoseEinstein condensates (BECs) confined by a parabolic potential.
The simulations emulate the motion of a localized laserbeam
spot through the trapped condensate.
Our results for the 1D case indicate that, due to
the inhomogeneity of the BEC density, the
critical speed for nucleation, as a function of the condensate density
displays two distinct dependences.
In particular, the squareroot of the critical density for
nucleation as a function of speed displays two different linear
regimes corresponding to small and large velocities.
Effectively, the emission of gray solitons
and vortexantivortex pairs occurs for any velocity of the
impurity, as any given velocity will be supercritical in a region
with a sufficiently small density. At longer times, the first
nucleation is followed by generation of an array of solitons in 1D
("soliton train") or vortex pairs in 2D
("vortex street") by the moving object.
[top]
Dynamics and Manipulation of MatterWave Solitons in Optical Superlattices.
M.A. Porter,
P.G. Kevrekidis,
R. CarreteroGonzález, and
D.J. Frantzeskakis.
Phys. Lett. A
352 (2006) 210.
PDF.
We consider matterwave solitons of the bright,
dark and gap type in optical superlattices. We analyze the existence
and stability properties of such coherent structures. We then
use these properties to illustrate that
(timedependent) "dynamical superlattices" is an ideal setting
for the deposition, guidance and,
generally, manipulation of these solitons. Such experimentally
accessible protocols may pave the way for the controllable use of
solitonic quantum "bits".
[top]
Vector solitons with an embedded domain wall.
P.G. Kevrekidis,
H. Susanto,
R. CarreteroGonzález,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. E 72 (2005) 066604.
PDF.
We present a class of soliton solutions to a system of two coupled nonlinear
Schrödinger equations, with an intrinsic domain wall (DW) which
separates regions occupied by two different fields. The model describes a
binary mixture of two BoseEinstein condensates (BECs) with interspecies
repulsion. For the attractive/repulsive interactions inside each species, we
find solutions which are bright/dark solitons in each component, while for
the opposite signs of the intraspecies interaction, a brightdark soliton
pair is found (each time, with the intrinsic DW). These solutions can arise
in the context of discrete lattices, and most of them can be supported in
continuum settings by an external parabolic trap. The stability of the
solitons with intrinsic DWs is examined, and the evolution of unstable ones
is analyzed. We also briefly discuss the possibility of generating such
families of solutions in the presence of linear coupling between the
components, and an application of the model to bimodal light propagation in
nonlinear optics.
[top]
Discrete Solitons and Vortices on Anisotropic Lattices.
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
B.A. Malomed,
and A.R. Bishop.
Phys. Rev. E 72 (2005) 046613.
PDF.
We consider effects of anisotropy on solitons of various types in
twodimensional nonlinear lattices, using the discrete nonlinear
Schrödinger equation as a paradigm model. For fundamental
solitons, we develop a variational approximation, which predicts
that broad quasicontinuum solitons are unstable, while their
strongly anisotropic counterparts are stable. By means of
numerical methods, it is found that, in the general case, the
fundamental solitons and simplest onsitecentered vortical
solitons ("vortex crosses") feature enhanced or reduced stability
areas, depending on the strength of the anisotropy. More
surprising is the effect of anisotropy on the socalled
"supersymmetric" intersitecentered vortices ("vortex
squares"), with the topological charge S equal to the square's
size M (for S<=M), dynamical properties of vortex squares
are not qualitatively different from those of the fundamental
soliton and vortex crosses): we predict in an analytical form by
means of the LyapunovSchmidt theory, and confirm by numerical
results, that arbitrarily weak anisotropy results in dramatic
changes in the stability and dynamics in comparison with the
degenerate, in this case, isotropic limit.
[top]
Multistable Solitons of the CubicQuintic Discrete Nonlinear Schrödinger Equation.
R. CarreteroGonzález,
J.D. Talley,
C. Chong, and
B.A. Malomed.
Physica D
216 (2006) 7789.
PDF.
We analyze the existence and stability of localized solutions in
the onedimensional discrete nonlinear Schrödinger (DNLS)
equation with a combination of selffocusing cubic and defocusing
quintic onsite nonlinearities. We provide a stability diagram for
different families of soliton solutions, that suggests the
(co)existence of infinitely many branches of stable localized
solutions. Bifurcations which occur with the
increase of the coupling constant are studied in a numerical form,
and a variational approximation is developed for accurate
prediction of the principal saddlenode bifurcation. Salient
properties of the model, which distinguish it from the wellknown
cubic DNLS equation, are the existence of two different types of
symmetric solitons, and, especially, stable asymmetric
soliton solutions that are found in narrow regions of the
parameter space. The asymmetric solutions appear from and
disappear back into the symmetric ones via loops of forward and
backward pitchfork bifurcations.
[top]
Trapped bright solitons in the presence of localized inhomogeneities.
G. Herring,
P.G. Kevrekidis,
R. CarreteroGonzález,
B.A. Malomed,
D.J. Frantzeskakis,
and A.R. Bishop.
Phys. Lett. A 345 (2005) 144.
PDF.
We examine the dynamics of a bright solitary wave in the presence of
a repulsive or attractive localized "impurity" in BoseEinstein
condensates (BECs). We study the generation and stability of a
pair of steady states in the vicinity of the impurity as the
impurity strength is varied. These two new steady
states, one stable and one unstable, disappear through a
saddlenode bifurcation as the strength of the impurity is
decreased. The dynamics of the soliton is also examined in all the
cases (including cases where the soliton is offset from one of
the relevant fixed points). The numerical results are corroborated
by theoretical calculations which are in very good agreement with
the numerical findings.
[top]
ThreeDimensional Nonlinear Lattices:
From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes.
R. CarreteroGonzález,
P.G. Kevrekidis,
B.A. Malomed, and
D.J. Frantzeskakis.
Phys. Rev. Lett. 94 (2005) 203901.
PDF.
We construct a variety of novel localized
topological structures in the 3D discrete nonlinear
Schrödinger equation. The states can be created in
BoseEinstein condensates trapped in strong optical lattices, and
crystals built of microresonators. These new structures, most of
which have no counterparts in lower dimensions, range from
multipole patterns and diagonal vortices to
vortex "cubes" (stack of two quasiplanar vortices)
and "diamonds" (formed by two orthogonal vortices).
[top]
Multifrequency Synthesis Using two Coupled Nonlinear Oscillator Arrays.
A. Palacios,
R. CarreteroGonzález,
P. Longhini,
N. Renz,
V. In,
A. Kho,
B. Meadows, and
J. Neff.
Phys. Rev. E 72 (2005) 026211.
PDF.
A scheme that exploits the theory of symmetrybreaking
bifurcations for generating a spatiotemporal pattern in which
one of two interconnected arrays, each with N Van der Pol
oscillators, oscillates at N times the frequency of the other is
illustrated. A bifurcation analysis demonstrates that this type of
frequency generation cannot be realized without the mutual
interaction between the two arrays. It is also demonstrated that
the mechanism for generating these frequencies between the
two arrays is different from that of a masterslave interaction, a
synchronization effect, or that of subharmonic and
ultraharmonic solutions generated by forced systems. This
kind of frequency generation scheme can find applications in
the newly developed field of nonlinear antenna and radar
systems.
[top]
Vortices in BoseEinstein Condensates: Some Recent Developments.
P.G. Kevrekidis,
R. CarreteroGonzález,
D.J. Frantzeskakis, and
I.G. Kevrekidis.
Mod. Phys. Lett. B,
18 (2004) 14811505.
PDF.
In this brief review, we summarize a number of recent developments
in the study of vortices in BoseEinstein condensates, a topic of
considerable theoretical and experimental interest in the past few
years. We examine the generation of vortices, by means of phase
imprinting, as well as via dynamical instabilities. Their stability
is subsequently examined in the presence of purely magnetic trapping,
and in the combined presence of magnetic and optical trapping.
We then study pairs of vortices and their interactions, illustrating
a reduced description in terms of ordinary differential equations for
the vortex centers. In the realm of two vortices, we also consider
the existence of stable dipole clusters for
twocomponent condensates. Last but not least, we discuss clusters of
vortices, the socalled vortex lattices and analyze some of their
intriguing dynamical features. A number of interesting future directions
are highlighted.
[top]
Nonlinear Lattice Dynamics of BoseEinstein Condensates.
M.A. Porter,
R. CarreteroGonzález,
P.G. Kevrekidis and
B.A. Malomed,
Chaos, 15 (2005) 015115.
PDF.
Selected for the Virtual Journal of Biological Physics Research volume 9, issue 7 (2005).
The FermiPastaUlam (FPU) model, which was proposed 50 years ago
to examine thermalization in nonmetallic solids and develop
"experimental" techniques for studying nonlinear problems,
continues to yield a wealth of results in the theory and
applications of nonlinear Hamiltonian systems with many degrees of
freedom. Inspired by the studies of this seminal model,
solitarywave dynamics in lattice dynamical systems have proven
vitally important in a diverse range of physical
problemsincluding energy relaxation in solids, denaturation of
the DNA double strand, selftrapping of light in arrays of
optical waveguides, and BoseEinstein condensates (BECs) in
optical lattices. BECS, in particular, due to their widely ranging and
easily manipulated dynamical apparatuseswith one to three
spatial dimensions, positivetonegative tuning of the
nonlinearity, one to multiple components, and numerous
experimentally accessible external trapping potentialsprovide one
of the most fertile grounds for the analysis of
solitary waves and their interactions. In this paper, we review
recent research on BECs in the presence of deep periodic potentials, which
can be reduced to nonlinear chains in appropriate circumstances. These
reductions, in turn, exhibit many of the remarkable
nonlinear structures (including solitons, intrinsic localized
modes, and vortices) that lie at the heart of the nonlinear science
research seeded by the FPU paradigm.
[top]
Statics, Dynamics and Manipulation of Bright Matterwave Solitons in Optical Lattices.
P.G. Kevrekidis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
B.A. Malomed,
G. Herring and
A.R. Bishop.
Phys. Rev. A, 71 (2005) 023614.
PDF.
In this work, motivated by the recent experimental developments in the
context of BoseEinstein condensates, we consider a bright soliton in
the presence of a
parabolic magnetic trap and a periodic optical lattice.
We examine the steady states in the presence of such a potential
using the results of LyapunovSchmidt theory, as well
as identify their linear stability, and find good agreement with full
numerical calculations. We then proceed to use the optical lattice
in a dynamical way as a mean of trapping (i.e., stopping),
moving (i.e., displacing) and guiding the solitary waves in a prescribed way.
We also briefly discuss the emission of sound waves for a moving
soliton in the presence of the combined magnetic trap and optical lattice potential.
[top]
HigherOrder Vortices in Nonlinear Dynamical Lattices.
P.G. Kevrekidis,
B.A. Malomed,
D.J. Frantzeskakis, and
R. CarreteroGonzález.
Focus on Soliton Research,
Editor L.V. Chen, Nova
Science Publishers (2006) 139166.
PDF.
In this paper, we investigate localized discrete states with a nonzero topological
charge (discrete vortices) in the prototypical model of dynamicallattice
systems, based on the two and threedimensional (2D and 3D) discrete nonlinear
Schrödinger (DNLS) equation, with both attractive and repulsive onsite
cubic nonlinearity. Systems of two nonlinearly coupled DNLS equations are considered
too. We report new results concerning the existence and, especially, stability
of the vortices with higher values of the topological charge S (S=2,3,4).
Quasivortices, i.e., stable solutions of the quadrupole and octupole types,
which replace unstable vortices with S=2 and 4, respectively, are also
found. The vortices of the gapsoliton type, which are found in the defocusing
(repulsive) model, are quite different, as concerns the stability, from their
counterparts in the focusing (attractive) models. In the twocomponent system,
stable compound vortices of the type (S_{1},S_{2})
=(1,ą1) are found, the stability area being larger
for the (+S,S) species. In the 3D case, besides finding stable
vortices with S=1 and 3, a novel possibility is reported, viz., a stable
twocomponent complex with mutually orthogonal vortices in the components. Applications
of the results to nonlinear optics optics and BoseEinstein condensates are
briefly discussed.
[top]
Controlling the motion of dark solitons by means of periodic potentials:
Application to BoseEinstein condensates in optical lattices.
G. Theocharis,
D.J. Frantzeskakis,
R. CarreteroGonzález,
P.G. Kevrekidis and
B.A. Malomed.
Phys. Rev. E 71 017602 (2005) 017602.
PDF.
We demonstrate that the motion of dark solitons (DSs) can be controlled by
means of periodic potentials. The mechanism is realized in terms of
cigarshaped BoseEinstein condensates confined in a harmonic magnetic
potential, in the presence of an optical lattice (OL). In the case when the
OL period is comparable to the width of the DS, we demonstrate that (a) a
moving dark soliton can be captured, switching on the OL; and (b) a
stationary DS can be dragged by a moving OL.
[top]
Threedimensional solitary waves and vortices in a discrete nonlinear Schrödinger
lattice.
P.G.
Kevrekidis, B.A. Malomed, D.J.
Frantzeskakis and R.
CarreteroGonzález.
Phys. Rev. Lett.
93 (2004) 080403.
PDF.
In a benchmark dynamicallattice model in three dimensions, the discrete
nonlinear Schrödinger equation, we find vortex solitons with various
values of the topological charge S.
Stability regions for the vortices with S = 0,1,3 are investigated. The
S = 2 vortex is unstable, spontaneously rearranging into a stable one
with S = 3. In a twocomponent extension of the model, we
find a novel class of stable structures, consisting of vortices in the different
components, perpendicularly oriented to each other. Selflocalized states of the
proposed types can be observed experimentally in BoseEinstein condensates
trapped in optical lattices, and in photonic crystals built of microresonators.
[top]
Domain Walls of SingleComponent BoseEinstein Condensates in External Potentials.
P.G.
Kevrekidis, B.A. Malomed, D.J.
Frantzeskakis,
A.R. Bishop,
H.E. Nistazakis and
R.
CarreteroGonzález.
Math. Comput. Simulat., 69 (2005) 334345.
PDF.
We demonstrate the possibility of creating domain walls described by a single
component Gross Pitaevskii equation with attractive interaction, in the presence
of an opticallattice potential. While it is found that the extended domain
wall is unstable, we show that the external magnetic trap can stabilize it.
Stable solutions include twisted domain walls, as well as asymmetric solitons.
The results also apply to spatial solitons in planar waveguides with transverse
modulation of the refractive index.
[top]
A ParrinelloRahman Approach to Vortex Lattices.
R.
CarreteroGonzález, P.G.
Kevrekidis, I.G.
Kevrekidis, D.
Maroudas and D.J.
Frantzeskakis.
Phys. Lett. A
341 (2005) 128134.
PDF.
We present a framework for studying vortex lattice patterns and their structural
transitions, using the ParrinelloRahman (PR) method for MolecularDynamics
(MD) simulations. Assuming an interaction between vortices derived from a GinzburgLandau
fieldtheoretic context, we extract the groundstate of a "vortex gas''
using the PRMD technique and find it to be a triangular pattern. Other patterns
are also obtained for special initial conditions. Generalizations of the technique,
such as the inclusion of external potentials or excitation of quadrupolar modes,
are also commented upon..
[top]
Precise computations of chemotactic collapse using moving mesh methods.
C.J.
Budd,
R. CarreteroGonzález and
R.D. Russell.
J. Comput. Phys., 202, 2 (2005) 463487.
PDF.
We study the application of dynamic (scaleinvariant) remeshing techniques to
the problem of chemotactic collapse. These methods are based on moving a spatial
mesh to equidistribute a carefully chosen monitor function. It is shown that
the results of these computations are fully consistent with the asymptotic description
of the collapse phenomenon given by Herrero and Velázquez (Math. Ann.,
306 (1996), 583623).
[top]
Families of matterwaves for TwoComponent BoseEinstein Condensates.
P.G.
Kevrekidis, G.
Theocharis, D.J.
Frantzeskakis, B.A.
Malomed and R.
CarreteroGonzález.
Eur.
Phys. J. D: At. Mol. Opt. Phys., 28,
2, (2004) 181185. PDF.
We produce several families of solutions for twocomponent nonlinear Schrödinger/GrossPitaevskii
equations. These include domain walls and the first example of an antidark or
gray soliton in the one component, bound to a bright or dark soliton in the
other. Most of these solutions are linearly stable in their entire domain of
existence. Some of them are relevant to nonlinear optics, and all to BoseEinstein
condensates (BECs). In the latter context, we demonstrate robustness of the
structures in the presence of parabolic and periodic potentials (corresponding,
respectively, to the magnetic trap and optical lattices in BECs).
[top]
Dark soliton dynamics in spatially inhomogeneous media: Application to BoseEinstein
condensates.
G.
Theocharis, D.J.
Frantzeskakis, P.G.
Kevrekidis, R.
CarreteroGonzález and B.A.
Malomed.
Math.
Comput. Simulat. 69 (2005) 537552.
PDF.
We study the dynamics of dark solitons in spatially inhomogeneous media with
app lications to cigarshaped BoseEinstein condensates trapped in a harmonic
magnet ic potential and a periodic potential representing an optical lattice.
We distin guish and systematically investigate the cases with the optical lattice
period being smaller, larger, or comparable to the width of the dark soliton.
Analytical results, based on perturbation techniques, for the motion of the
dark soliton are obtained and compared to direct numerical simulations. Radiation
eff ects are also considered. Finally, we demonstrate that a moving optical
lattice may capture and drag a dark soliton.
[top]
Vortices in a BoseEinstein condensate confined by an optical lattice.
P.G.
Kevrekidis, R.
CarreteroGonzález, G.
Theocharis, D.J.
Frantzeskakis and B.A.
Malomed.
J.
Phys. B: At. Mol. Phys. 36
(2003) 34673476. PDF.
We investigate dynamics of vortices in repulsive BoseEinstein condensates
in the presence of an optical lattice (OL) and a parabolic magnetic trap. The
dynamics is sensitive to the phase of the OL potential relative to the magnetic
trap, and depends less on the OL strength. For the cosinusoidal OL potential,
a local minimum is generated at the trap's center, creating a stable equilibrium
of the vortex, while in the case of the sinusoidal potential, the vortex is
expelled from the center, demonstrating spiral motion. Cases when the vortex
is created far from the trap's center are also studied, revealing slow outward
drift. Numerical results are explained in an analytical form by means of a variational
approximation. Finally, motivated by a discrete model (which is tantamount to
the case of the strong OL lattice), we present a novel type of a vortex, consisting
of two pairs of antiphase solitons.
[top]
Stability of dark solitons in a BoseEinstein condensate confined in an optical
lattice.
P.G.
Kevrekidis,
R.
CarreteroGonzález,
G.
Theocharis,
D.J.
Frantzeskakis and
B.A.
Malomed.
Phys.
Rev. A,
68
035602 (2003).
PDF, [
ERRATUM].
We investigate the stability of dark solitons (DSs) in an effectively onedimensional
BoseEinstein condensate in the presence of the magnetic parabolic trap and
an optical lattice (OL). The analysis is based on both the full GrossPitaevskii
equation and its tightbinding approximation counterpart (discrete nonlinear
Schrödinger equation). We find that most DSs are subject to weak instabilities,
although continuum DSs may sometimes be stable, which is determined by the
OL period and amplitude. The instability, if present, sets in at large times
and is characterized by quasiperiodic oscillations of the DS about the minimum
of the parabolic trap.
[top]
Variational Mesh Adaptation Methods for Axisymmetrical Problems with Applications
to Blowup.
W. Cao,
R. CarreteroGonzález,
W. Huang and
R.D. Russell.
SIAM
Journal on Numerical Analysis 41,1
(2003) 235257. PDF.
We study variational mesh adaptation for axially symmetric solutions to two dimensional
problems. The study is focused on the relationship between the mesh density distribution
and the monitor function and is carried out for a traditional functional that
includes several widely used variational methods as special cases and a recently
proposed functional that allows for a weighting between mesh isotropy (or regularity)
and global equidistribution of the monitor function. The main results are stated
in Theorems 4.1 and 4.2. For axially symmetric problems, it is natural to choose
axially symmetric mesh adaptation. This requires that the monitor function be
chosen in the form
G=λ_{1}(r)e_{r}e_{r}^{T} +
λ_{2}(r)e_{θ}e_{θ}^{T} +
where e_{r} and e_{θ} are the radial and angular unit vectors.
It is shown that when higher mesh concentration at the origin is desired,
a choice of λ_{1} and λ_{2} satisfying λ_{1}(0) < λ_{2}(0) will make the mesh denser
at r=0 than in the surrounding area whether or not λ_{1} has a maximum value at r=0.
The purpose can also be served by choosing λ_{1} to have a local maximum at r=0 when a Winslowtype monitor function
with λ_{1}(r)=λ_{2}(r) is employed. On the other hand,
it is shown that the traditional functional provides little control over mesh
concentration around a ring r=r_{λ}>0 by choosing
λ_{1} and λ_{2}.
In contrast, numerical results show that the new functional provides better
control of the mesh concentration through the monitor function. Twodimensional
numerical results are presented to support the analysis.
[top]
Localized breathing oscillations for BoseEinstein condensates in periodic
traps.
R. CarreteroGonzález and
K. Promislow.
Phys.
Rev. A
66, 3,
033610 (2002). PDF.
We demonstrate the existence of localized oscillatory breathers for quasionedimensional
BoseEinstein condensates confined in periodic potentials. The breathing behavior
corresponds to positionoscillations of individual condensates about the minima
of the potential lattice. Localized oscillations are identified with homoclinic
tangles of a reduced twodimensional map on the oscillation amplitudes. We deduce
the structural stability of the localized oscillations from the construction.
The stability is confirmed numerically for perturbations to the initial state
of the condensate, to the potential trap, as well as for external noise. We
also construct periodic and chaotic extended oscillations for the chain of condensates.
All our findings are verified by direct numerical integration of the GrossPitaevskii
equation in one dimension.
[top]
Stability of attractive BoseEinstein condensates in a periodic potential.
J.C. Bronski,
L.D. Carr,
R. CarreteroGonzález,
B. Deconinck,
J.N. Kutz and
K. Promislow.
Phys. Rev. E
64, 5,
056615 (2001). PDF.
The cubic nonlinear Schrödinger equation with repulsive nonlinearity
and an elliptic function potential models a quasionedimensional repulsive
dilute gas BoseEinstein condensate trapped in a standing light wave. New families
of stationary solutions are presented. Some of these solutions have neither
an analog in the linear Schrödinger equation nor in the integrable nonlinear
Schrödinger equation. Their stability is examined using analytic and numerical
methods. All trivialphase stable solutions are deformations of the ground state
of the linear Schrödinger equation. Our results show that a large number
of condensed atoms is sufficient to form a stable, periodic condensate. Physically,
this implies stability of states near the ThomasFermi limit.
[top]
Modelling desert dune fields based on discrete dynamics.
H. Momiji,
S.R. Bishop,
R. CarreteroGonzález and
A. Warren.
Discrete Dynamics in Nature and Society, 7,
1 (2002)
717. PDF.
Two mathematical models to simulate desert dune fields based on discrete dynamics
are proposed. Both models are developed by simplifying two mechanisms; sand
transport induced by wind and gravitational shaping. Despite complex wind flow
over desert dune fields some recurrent, spatial features occur. These models
comprise our understanding of the effect of various mechanisms and succeeded
in vividly demonstrating some important aspects of dune field. For model validation,
comparison between simulated results and nature is necessary on different length
scales.
[top]
Simulation of the effect of wind speedup in the formation of transverse dune
fields.
H. Momiji,
R. CarreteroGonzález,
S.R. Bishop and
A. Warren.
Earth Surface Processes and Landforms,
25, 8 (2000) 905918. PDF.
A computer simulation model for transversedunefield dynamics, corresponding
to a unidirectional wind regime, is developed. In a previous formulation, two
distinct problems were found regarding the crosssectional dune shape, namely
the erosion in the lee of dunes and the steepness of the windward slopes. The
first problem is solved by introducing no erosion in shadow zones. The second
issue is overcome by introducing a wind speedup (shear velocity increase) factor,
which can be accounted for by adding a term to the original transport length,
which is proportional to the surface height. By incorporating these features
we are able to model dunes whose individual shape and collective patterns are
similar to those observed in nature. Moreover we show how the introduction of
a nonlinear shearvelocityincrease term leads to the reduction of dune height,
and this may result in an equilibrium dune field configuration. This is thought
to be because the nonlinear increase of the transport length makes the sand
trapping efficiency lower than unity, even for higher dunes, so that the incoming
and the outgoing sand flux are in balance. To fully describe the interdune
morphology more precise dynamics in the lee of the dune must be incorporated.
[top]
Quasidiagonal approach to the estimation of Lyapunov spectra for spatiotemporal
systems from multivariate time series.
R. CarreteroGonzález,
S. Řrstavik and
J. Stark.
Phys. Rev. E
62, 5 (2000)
64296439. PDF.
We describe methods of estimating the entire Lyapunov spectrum of a spatially
extended system from multivariate timeseries observations. Provided that the
coupling in the system is short range, the Jacobian has a banded structure and
can be estimated using spatially localised reconstructions in low embedding
dimensions. This circumvents the "curse of dimensionality" that prevents the
accurate reconstruction of highdimensional dynamics from observed time series.
The technique is illustrated using coupled map lattices as prototype models
for spatiotemporal chaos and is found to work even when the coupling is not
strictly local but only exponentially decaying.
[top]
Thermodynamic limit from small lattices of coupled maps.
R. CarreteroGonzález,
S. Řrstavik,
J. Huke,
D.S. Broomhead and
J. Stark.
Phys. Rev. Lett.
83, 18 (1999) 36333636. PDF.
We compare the behaviour of a small truncated coupled map lattice with random
inputs at the boundaries with that of a large deterministic lattice essentially
at the thermodynamic limit. We find exponential convergence for the probability
density, predictability, power spectrum, and twopoint correlation with increasing
truncated lattice size. This suggests that spatiotemporal embedding techniques
using local observations cannot detect the presence of spatial extent in such
systems and hence they may equally well be modelled by a local low dimensional
stochastically driven system.
[top]
Estimation of intensive quantities in spatiotemporal systems from timeseries.
S. Řrstavik,
R. CarreteroGonzález and
J. Stark.
Physica D
147 (2000) 204220.
PDF.
We study multivariate timeseries generated by coupled map lattices exhibiting
spatiotemporal chaos and investigate to what extent we are able to estimate
various ntensive measures of the underlying system without explicit knowledge
of the system dynamics. Using the rescaling and interleaving properties of the
Lyapunov spectrum of systems in a spatiotemporally chaotic regime and paying
careful attention to errors introduced by subsystem boundary effects, we develop
algorithms that are capable of estimating the Lyapunov spectrum from time series.
We analyse the performance of these and find that the choice of basis used to
fit the dynamics is crucial: when the local dynamics at a lattice site is well
approximated by this basis we are able to accurately determine the full Lyapunov
spectrum. However, as the local dynamics moves away from the space spanned by
this basis the performance of our algorithm deteriorates.
[top]
Scaling and interleaving of subsystem Lyapunov exponents for spatiotemporal
systems.
R. CarreteroGonzález,
S. Řrstavik and
J. Stark.
Chaos
9,
2 (1999)
466482 . PDF.
The computation of the entire Lyapunov spectrum for extended dynamical systems
is a very time consuming task. If the system is in a chaotic spatiotemporal
regime it is possible to approximately reconstruct the Lyapunov spectrum from
the spectrum of a subsystem in a very cost effective way. In this work we present
a new rescaling method, which gives a significantly better fit to the original
Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous
evolution in a onedimensional coupled map lattice but appears to be equally
valid in a much wider range of cases. We evaluate the performance of our rescaling
method by comparing it to the conventional rescaling (dividing by the relative
subsystem volume) for one and twodimensional lattices in spatiotemporal chaotic
regimes. In doing so we notice that the Lyapunov spectra for consecutive subsystem
sizes are interleaved and we discuss the possible ways in which this may arise.
Finally, we use the new rescaling to approximate quantities derived from the
Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and KolmogorovSinai
entropy) finding better convergence as the subsystem size is increased than
with conventional rescaling.
[top]
Onedimensional dynamics for travelling fronts in coupled map lattices.
R. CarreteroGonzález,
D.K. Arrowsmith and
F. Vivaldi.
Phys. Rev. E
61, 2 (2000)
13291336. PDF.
Multistable coupled map lattices typically support travelling fronts, separating
two adjacent stable phases. We show how the existence of an invariant function
describing the front profile, allows a reduction of the infinitelydimensional
dynamics to a onedimensional circle homeomorphism, whose rotation number gives
the propagation velocity. The modelocking of the velocity with respect to the
system parameters then typically follows. We study the behaviour of fronts near
the boundary of parametric stability, and we explain how the modelocking tends
to disappear as we approach the continuum limit of an infinite density of sites.
[top]
Low dimensional travelling interfaces in coupled map lattices.
R. CarreteroGonzález.
Int. J. Bifurcation and Chaos
7, 12 (1997)
27452754. PDF.
We study the dynamics of the travelling interface arising from a bistable
piecewise linear oneway coupled map lattice. We show how the dynamics of the
interfacial sites, separating the two superstable phases of the local map, is
finite dimensional and equivalent to a toral map. The velocity of the travelling
interface corresponds to the rotation vector of the toral map. As a consequence,
a rational velocity of the travelling interface is subject to modelocking with
respect to the system parameters. We analytically compute the Arnold's tongues
where particular spatiotemporal periodic orbits exist. The boundaries of the
modelocked regions correspond to bordercollision bifurcations of the toral
map. By varying the system parameters it is possible to increase the number
of interfacial sites corresponding to a bordercollision bifurcation of the
interfacial attracting cycle. We finally give some generalizations towards smooth
coupled map lattices whose interface dynamics is typically infinite dimensional
[top]
ModeLocking in Coupled Map Lattices.
R. CarreteroGonzález,
D.K. Arrowsmith and
F. Vivaldi.
Physica D
103, 1/4 (1997) 381403.
PDF.
We study propagation of pulses along oneway coupled map lattices, which originate
from the transition between two superstable states of the local map. The velocity
of the pulses exhibits a staircaselike behaviour as the coupling parameter
is varied. For a piecewise linear local map, we prove that the velocity of
the wave has a Devil's staircase dependence on the coupling parameter. A wave
travelling with rational velocity is found to be stable to parametric perturbations
in a manner akin to rational modelocking for circle maps. We provide evidence
that modelocking is also present for a broader range of maps and couplings.
[top]
Regular and Chaotic Behaviour in an Extensible Pendulum.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Eur. J. Phys.
15, 3 (1994)
139148. PDF.
The extensible pendulum is studied numerically to illustrate the Hamiltonian
transition to chaos. This is an apparently simple system which is well suited
to explain concepts related with the onset of chaos. Using Poincaré sections
we exhibit the lowenergy regular motion and the coexistence of stochastic and
regular motion at intermediate energies. We employ other diagnostic techniques
for checking our conclusions.
[top]
Evidence of Chaotic Behaviour in JordanBransDicke Cosmology.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Phys. Lett. A 188, 1 (1994) 4854.
PDF.
We study numerically the properties of solutions of spatially homogeneous
Bianchitype IX cosmological models in the JordanBransDicke theory of gravitation.
Solutions are obtained in which the scale factors undergo irregular oscillations.
The estimate of the maximum Lyapunov exponent is found to be positive in the
cases studied. These results seem to be the first pieces of evidence in the
literature off chaotic behaviour in JordanBransDicke cosmology. The range
of values of the JordanBransDicke coupling parameter considered is (500,1).
[top]
Energy localization and transport in twodimensional electrical lattices.
L.Q. English,
F. Palmero,
J. Stormes,
J. Cuevas,
R. CarreteroGonzález, and
P.G. Kevrekidis.
2013 Int. Symposium on Nonlinear Theory & Its Applications, Santa Fe, New Mexico, USA, September 812, 2013.
PDF.
Intrinsic localized modes (ILMs) have been generated and characterized in
twodimensional nonlinear electrical lattices which were driven by a
spatiallyuniform voltage signal. These ILMs were found to be either
stationary or mobile, depending on the details of the lattice unitcell,
as had already been reported in onedimensional lattices; however, the
motion of these ILMs is qualitatively different in that it lacks a
consistent direction. Furthermore, the hopping speed seems to be somewhat
reduced in two dimensions due to an enhanced PeierlsNabarro (PN)barrier.
We investigate both square and honeycomb lattices composed of 6x6
elements. These direct observations were further supported by numerical
simulations based on realistic models of circuit components. The numerical
study moreover allowed for an analysis of ILM dynamics and pattern formation
for larger lattice sizes.
[top]
Optical Manipulation of Matter Waves.
R. CarreteroGonzález,
P.G. Kevrekidis,
D.J. Frantzeskakis, and
B.A. Malomed.
Proc. SPIE Int. Soc. Opt. Eng. 5930 (2005) 59300L.
PDF.
Recent experimental and theoretical progress in the studies of BoseEinstein
condensation (BEC) has precipitated an intense effort to understand and
control interactions of nonlinear matterwaves. Key ingredients in
manipulations of matterwaves in BECs are a) external localized impurities
(generated by focused laser beams) and b) periodic potentials (generated by
interference patterns from multiple laser beams illuminating the
condensate). In this work we demonstrate the ability of timedependent
external optical potentials to drag, capture and pin a wide range of
localized BEC states, such as dark and bright solitons. The stability and
existence of pinned states is analyzed using perturbation techniques, which
predict results that are well corroborated by direct numerical simulations.
The control of these macroscopic quantum states has important applications
in the realm of quantum storage and processing of information, with
potential implications for the design of quantum computers.
[top]
Multifrequency Pattern Generation Using GroupSymmetric Circuits.
J. Neff,
V. In,
B. Meadows,
C. Obra,
A. Palacios, and
R. CarreteroGonzález,
2006 IEEE International Symposium on Circuits and Systems.
PDF.
This paper explores the use of networked electronic circuits, which
have symmetrical properties, for generating patterns with multiple
frequencies. Using a simple bistable subcircuit, connected in a
network with a specific topology, the principal operating frequencies
of the network are divided in to two groups, with one group oscillating
at twice the frequency of the other group. Specifically, grouptheoretic
arguments are used to dictate the particular coupling topology between
the unit bistable cell. These concepts are demonstrated in a simple
and compact CMOS circuit. The circuit is minimalistic, and demonstrates
how simple and robust circuits can be used to generate useful patterns.
[top]
The Curvature Criterion and the Dynamics of a Rolling Elastic Cylinder.
M. Arizmendi,
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Advanced Series in Nonlinear Dynamics, Vol. 8. New Trends for Hamiltonian
Systems and Celestial Mechanics.
E.A.Lacomba & J.Llibre (Eds.). Wolrd Scientific, July (1996) 113.
The motion of an expanding and contracting cylinder rolling on a curved surface
is analysed. We establish the existence of autoparametric instabilities for
certain values of the cylinder parameters and also found its integrable cases.
We next use what we have called the curvature criterion to predict an orderchaosorder
transition whenever the system is chaotic. The Poincaré surface of section
method is used to check some of the predictions with good results. The curvature
criterion and some of its implications are also briefly discussed.
[top]
Nonlinear behaviour in the JBD scalartensor
theory.
R. CarreteroGonzález,
P. Chauvet,
H.N. NúńezYépez and
A.L. SalasBrito.
Proceedings of the "Int. Conference on Aspects of General Relativity and Mathematical
Physics". Mexico City, June 24, 1993.
N.Bretón, R.Capovilla & T.Matos (Eds). CINVESTAV, Mexico City, (1994) 204209.
We apply techniques of nonlinear dynamics to a cosmological problem in the
Jordan BransDicke theory. The solutions presented here show irregular oscillatory
behaviour in the scale factors and a positive Liapunov exponent in the Bianchi
IX model. This is evidence of stochastic behaviour in the model.
[top]
Chaotic behaviour in JBD cosmology.
R. CarreteroGonzález,
H.N. NúńezYépez and
A.L. SalasBrito.
Proceedings of the "8th Latin American Symposium on Relativity and Gravitation",
Aguas de Lindoia, Brazil July 2530, 1993.
Gravitation: The Spacetime Structure.
P.S.Letelier & W.A.Rodrigues (Eds.). Wolrd Scientific, July (1994) 457461.
Solutions for the spatially homogeneous Bianchitype IX cosmological model
are studied in the Jordan BransDicke theory of gravitation. We find solutions
that seem to undergo irregular oscillations. This qualitative assessment of
the behaviour is corroborated by computing the maximum Liapunov exponent associated
with the solutions. We have found evidence for chaotic behaviour in the JBD
theory similar to the evidence for chaos found for the BianchiIX model in general
relativity. It seems as if the main contribution to stochasticity came from
the oscillating approach to the singularity.
[top]
Front propagation and modelocking in coupled map lattices.
R. CarreteroGonzález.
Ph.D. thesis, Dep. of Mathematical Sciences,
Queen Mary and Westfield College, London, UK, August 1997.
Location at Queen Mary
and Westfield College Library,
PDF,
Table of contents.
We study the propagation of coherent
signals through bistable oneway and diffusive coupled map lattices (CML). We
describe a simple mechanism that allows interfaces to travel along the lattice,
without damping or dispersion. This mechanism relies on a nondecreasing bistable
local map with two stable fixed points. The state of the lattice is then set
as a step state between the stable points and it is seen to advance along the
lattice with a welldefined velocity that depends on the coupling parameter
ε. For some local maps the velocity is shown to have εintervals where
it is modelocked to a rational value.
In order to understand the modelocking
phenomenon we introduce a continuous piecewise linear local map. We show how
the dynamics of the whole lattice (infinite system) may be reduced to a onedimensional
auxiliary map. The auxiliary map is a circlelike map whose rotation number
corresponds to the velocity of the travelling interface. We introduce symbolic
dynamics to fully understand the modelocking of the rotation number. We prove
that the velocity of the travelling interface has a Devil's staircase (a fractal
staircase) dependence on the coupling parameter. The Devil's staircase is modelocked
to rational plateaus and may be fully described via Farey sequences and modular
transformations.
Finally we give some numerical examples depicting modelocking of the velocity
for a wider range of couplings and local maps and we study the dependence of
plateau sizes on the coupling interaction range. The modelocking of the velocity
in CML allows an interface to travel at a constant speed despite parametric
perturbations giving structural stability to the front propagation and is present
in a very wide range of CMLs.
[top]
The transition to chaos on an extensible pendulum.
R. CarreteroGonzález.
B.Sc. Thesis, Facultad de Ciencias, Universidad Nacional Autónoma de México,
México, December 1992.
Extended version (in Spanish) of the paper Regular and Chaotic
Behaviour in an Extensible Pendulum. (see above).
[top]
Scitation Index
Math Reviews of some of my papers (requires MathScinet license)
You
can find some of my papers at the Los Alamos Eprint archive (arXiv.org)
My
papers in Physical Review journals (PRE, PRA & PRL)