## List of publications and abstracts

[Books], [Papers], [Proceedings], [Thesis].

### Papers

1. Dynamics of Few Co-rotating Vortices in Bose-Einstein Condensates.
R. Navarro, R. Carretero-González, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altuntaş, and D.S. Hall.
To appear in Phys. Rev. Lett., 2013. Abstract. PDF.
1. Inelastic Collisions of Solitary Waves in Anisotropic Bose-Einstein Condensates:
Sling-Shot Events and Expanding Collision Bubbles.

C. Becker, K. Sengstock, P. Schmelcher, R. Carretero-González, and P.G. Kevrekidis.
To be submitted, 2013. Abstract. PDF.
1. Nonlinear localized modes in two-dimensional electrical lattices.
L.Q. English, F. Palmero, J. Stormes, J. Cuevas, R. Carretero-González, and P.G. Kevrekidis.
Submitted, 2013. Abstract. PDF.
1. Solitons and their ghosts in PT-symmetric systems with defocusing nonlinearities.
V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González.
Submitted, 2012. Abstract. PDF.
1. Symmetry-breaking Effects for Polariton Condensates in Double-Well Potentials.
A.S. Rodrigues, P.G. Kevrekidis, J. Cuevas, R. Carretero-González, and D.J. Frantzeskakis.
Submitted 2012. Abstract. PDF.
1. Characteristics of Two-Dimensional 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. Carretero-González, P.G. Kevrekidis, M.J. Davis, and B.P. Anderson.
Submitted to Phys. Rev. Lett., 2012. Abstract. PDF. [Suplementary material].
1. Dark solitons and vortices in PT-symmetric nonlinear media:
from spontaneous symmetry breaking to nonlinear PT phase transitions.

V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González.
Phys. Rev. A 86 (2012) 013808. Abstract. PDF.
1. Vortices in Bose-Einstein Condensates: (Super)fluids with a twist.
P.G. Kevrekidis. R. Carretero-González, and D.J. Frantzeskakis.
Dynamical Systems Magazine, October 2011. Abstract. WWW.
1. A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and its Application to the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
Submitted to J. Comput. Appl. Math., 2012. Abstract. PDF.
1. A Two-Step High-Order Compact Scheme for the Laplacian Operator and its Implementation in an Explicit Method for Integrating the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
J. Comput. Appl. Math. 251 (2013) 33-46. Abstract. PDF.
1. Numerical Stability of Explicit Runge-Kutta Finite-Difference Schemes for the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
App. Num. Math. 71 (2013) 24-40. Abstract. PDF.
1. Dynamics of Vortex Dipoles in Confined Bose-Einstein Condensates.
P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, and D.S. Hall.
Phys. Lett. A 375 (2011) 3044-3050. Abstract. PDF.
1. Generation of localized modes in an electrical lattice using subharmonic driving.
L.Q. English, F. Palmero, P. Candiani, J. Cuevas, R. Carretero-González, P.G. Kevrekidis, and A.J. Sievers.
Phys. Rev. Lett. 108 (2012) 084101. Abstract. PDF.
1. Multiple dark-bright solitons in atomic Bose-Einstein condensates.
D. Yan, J.J. Chang, C. Hamner, P.G. Kevrekidis. P. Engels, V. Achilleos, D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
Phys. Rev. A 84 (2011) 053630. Abstract. PDF.
1. Discrete breathers in a nonlinear electric line: Modeling, Computation and Experiment.
F. Palmero, L.Q. English, J. Cuevas, R. Carretero-González, and P.G. Kevrekidis.
Phys. Rev. E 84 (2011) 026605. Abstract. PDF.
1. Guiding-Center Dynamics of Vortex Dipoles in Bose-Einstein Condensates.
S. Middelkamp, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, D.V. Freilich, and D.S. Hall.
Phys. Rev. A 84 (2011) 011605(R). Abstract. PDF.
1. Nonlinear Excitations, Stability Inversions and Dissipative Dynamics in Quasi-one-dimensional Polariton Condensates.
J. Cuevas, A.S. Rodrigues, R. Carretero-González, P.G. Kevrekidis, and D.J. Frantzeskakis.
Phys. Rev. B 83 (2011) 245140. Abstract. PDF.
1. Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings.
C. Chong, R. Carretero-González, B.A. Malomed, and P.G. Kevrekidis.
Physica D 240 (2011) 1205-1212. Abstract. PDF.
1. Controlling directed transport of matter-wave solitons using the ratchet effect.
M.A. Rietmann, R. Carretero-González, and R. Chacon.
Phys. Rev. A 83 (2011) 053617. Abstract. PDF.
1. Emergence and Stability of Vortex Clusters in Bose-Einstein Condensates: a Bifurcation Approach near the Linear Limit.
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
Physica D, 240 (2011) 1449-1459. Abstract. PDF.
1. Vortex Interaction Dynamics in Trapped Bose-Einstein Condensates.
P.J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, P.G. Kevrekidis, and D.J. Frantzeskakis.
Comm. Pure Appl. Ana. 10 (2011) 1589-1615. Abstract. PDF.
1. Dynamics of Dark-Bright Solitons in Cigar-Shaped Bose-Einstein Condensates.
S. Middelkamp, J.J. Chang, C. Hamner, R. Carretero-González, P.G. Kevrekidis, V. Achilleos, D.J. Frantzeskakis, P. Schmelcher, and P. Engels.
Phys. Lett. A 375 (2011) 642-646. Abstract. PDF.
Dark-bright (DB) oscillation movies:
[ Movie#1, Fig. 3 ]: Single DB for parameters in Nature Phys. 4, 496 (2008): ND=92,432 and NB=7,973, (fz,fy,fx)=(85,133,5.9) Hz,
[ Movie#2, Fig. 4.a ]: Single DB with bright soliton transverse dynamics: ND=88,181 and NB=1,058, (fz,fy,fx)=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.d ]: Two interacting DBs with out-of-phase (attractive) bright solitons: ND=5,243 and NB=817, (fz,fy,fx)=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.e ]: Two interacting DBs with in-phase (repulsive) bright solitons: ND=5,331 and NB=907, (fz,fy,fx)=(133,133,5.9) Hz.
1. Bifurcations, Stability and Dynamics of Multiple Matter-Wave Vortex States.
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, 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).
1. Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier.
Manjun Ma, R. Carretero-González, 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).
1. Existence, Stability, and Scattering of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation.
R.M. Caplan, R. Carretero-González. P.G. Kevrekidis, and B.A. Malomed.
Math. Comput. Simulat. 82 (2012) 1150-1171. Abstract. PDF.
1. Stability and dynamics of matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations.
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
J. Phys. B 43 (2010) 155303. Abstract. PDF.
1. Manipulation of Vortices by Localized Impurities in Bose-Einstein Condensates.
M.C. Davis, R. Carretero-González, 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).
1. Phase Separation and Dynamics of Two-component Bose-Einstein Condensates.
R. Navarro, R. Carretero-González, 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).
1. Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation.
R.M. Caplan, Q.E. Hoq, R. Carretero-González, and P.G. Kevrekidis.
Optics. Comm. 282 (2009) 1399-1405. Abstract. PDF.
1. Spinor Bose-Einstein condensate past an obstacle.
A.S. Rodrigues, P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, P. Schmelcher, T.J. Alexander, and. Yu.S. Kivshar.
Phys. Rev. A 79 (2009) 043603. Abstract. PDF.
1. Dissipative Solitary Waves in Periodic Granular Crystals.
R. Carretero-González, D. Khatri, M.A. Porter, P.G. Kevrekidis, and C. Daraio.
Phys. Rev. Lett. 102 (2009) 024102. Abstract. PDF.
1. Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice.
R. Chacon, D. Bote, and R. Carretero-González.
Phys. Rev. E 78 (2008) 036215. Abstract. PDF.
1. A Map Approach to Stationary Solutions of the Discrete Nonlinear Schrödinger Equation.
R. Carretero-González.
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.
1. Structure and stability of two-dimensional Bose-Einstein condensates under both harmonic and lattice confinement.
K.J.H. Law, P.G. Kevrekidis, B.P. Anderson, R. Carretero-González, and D.J. Frantzeskakis.
J. Phys. B, 41 (2008) 195303. Abstract. PDF.
1. Surface Solitons in Three Dimensions.
Q.E. Hoq, R. Carretero-González, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, Yu.V. Bludov, and V.V. Konotop.
Phys. Rev. E 78 (2008) 036605. Abstract. PDF.
1. Multistable Solitons in Higher-Dimensional Cubic-Quintic Nonlinear Schrödinger Lattices.
C. Chong, R. Carretero-González, B.A. Malomed, and P.G. Kevrekidis.
Physica D, 238 (2009) 126-136. Abstract. PDF.
1. Nonlinear dynamics of Bose-condensed gases by means of a q-Gaussian variational approach.
A.I. Nicolin and R. Carretero-González.
Physica A 387 (2008) 6032. Abstract. PDF.
1. Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity.
F. Palmero, R. Carretero-González, J. Cuevas, P.G. Kevrekidis, and W. Królikowski.
Phys. Rev. E 77 (2008) 036614. Abstract. PDF.
1. Dynamics of Vortex Formation in Merging Bose-Einstein Condensate Fragments.
R. Carretero-González, B.P. Anderson, P.G. Kevrekidis, D.J. Frantzeskakis, and C.N. Weiler.
Phys. Rev. A 77 (2008) 033625. Abstract. PDF.
1. Resonant energy transfer in Bose-Einstein condensates.
A.I. Nicolin, M.H. Jensen, J.W. Thomsen, and R. Carretero-González.
Physica D, 237 (2008) 2476-2481. Abstract. PDF.
1. Nonlinear Waves in Bose-Einstein Condensates:
Physical Relevance and Mathematical Techniques.

R. Carretero-González, D.J. Frantzeskakis, and P.G. Kevrekidis.
Nonlinearity 21 (2008) R139-R202. Abstract. PDF.
1. Radially Symmetric Nonlinear States of Harmonically Trapped Bose-Einstein Condensates.
G. Herring, L.D. Carr, R. Carretero-González, P.G. Kevrekidis, and D.J. Frantzeskakis.
Phys. Rev. A 77 (2008) 023625. Abstract. PDF.
1. Faraday waves in Bose-Einstein condensates.
A.I. Nicolin, R. Carretero-González, and P.G. Kevrekidis.
Phys. Rev. A 76 (2007) 063609. Abstract. PDF.
1. Extended Nonlinear Waves in Multidimensional Dynamical Lattices.
Q.E. Hoq, J. Gagnon, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and R. Carretero-González.
Math. Comput. Simulat., 80 (2009) 721-731. Abstract. PDF.
1. Polarized States and Domain Walls in Spinor Bose-Einstein Condensates.
H.E. Nistazakis, D.J. Frantzeskakis. P.G. Kevrekidis, B.A. Malomed, R. Carretero-González, and A.R. Bishop.
Phys. Rev. A 76 (2007) 063603. Abstract. PDF.
1. Bright-Dark Soliton Complexes in Spinor Bose-Einstein Condensates.
H.E. Nistazakis, D.J. Frantzeskakis. P.G. Kevrekidis, B.A. Malomed, and R. Carretero-González.
Phys. Rev. A 77 (2008) 033612. Abstract. PDF.
1. Symmetry breaking in linearly coupled dynamical lattices.
G. Herring, P.G. Kevrekidis. B.A. Malomed, R. Carretero-González, and D.J. Frantzeskakis.
Phys. Rev. E 76 (2007) 066606. Abstract. PDF.
1. Non-Equilibrium Dynamics and Superfluid Ring Excitations in Binary Bose-Einstein Condensates.
K.M. Mertes, J. Merrill, R. Carretero-González, D.J. Frantzeskakis, P.G. Kevrekidis, and D.S. Hall.
Phys. Rev. Lett. 99 (2007) 190402. Abstract. PDF.
Movies: [ cuts @ 60% and 30% for |1> and |2 > ], [ cuts @ 50% and 55% for |1> and |2 > ]
1. Čerenkov-like radiation in a binary superfluid flow past an obstacle.
H. Susanto, P.G. Kevrekidis. R. Carretero-González, B.A. Malomed, D.J. Frantzeskakis, and A.R. Bishop.
Phys. Rev. A 75 (2007) 055601. Abstract. PDF.
1. Vortex Structures Formed by the Interference of Sliced Condensates.
R. Carretero-González, N. Whitaker, P.G. Kevrekidis, and D.J. Frantzeskakis.
Phys. Rev. A, 77 (2008) 023605. Abstract. PDF.
1. Mode locking of a driven Bose-Einstein condensate.
A.I. Nicolin, M.H. Jensen, and R. Carretero-González.
Phys. Rev. E 75 (2007) 036208. Abstract. PDF.
1. Rotating matter waves in Bose-Einstein condensates.
T. Kapitula, P.G. Kevrekidis, and R. Carretero-González.
Physica D, 233 (2007) 112-137. Abstract. PDF.
1. Discrete surface solitons in two dimensions.
H. Susanto, P.G. Kevrekidis, B.A. Malomed, R. Carretero-González, and D.J. Frantzeskakis.
Phys. Rev. E 75 (2007) 056605. Abstract. PDF.
1. Mobility of Discrete Solitons in Quadratic Nonlinear Media.
H. Susanto, P.G. Kevrekidis, R. Carretero-González, B.A. Malomed, and D.J. Frantzeskakis.
Phys. Rev. Lett. 99 (2007) 214103. Abstract. PDF.
1. Skyrmion-like states in two- and three-dimensional dynamical lattices.
P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, B.A. Malomed, and F.K. Diakonos.
Phys. Rev. E 75 (2007) 026603. Abstract. PDF.
1. Multipole-mode solitons in Bessel optical lattices.
Y.V. Kartashov, R. Carretero-González, B.A. Malomed, V.A. Vysloukh, and Ll. Torner.
Optics Express 13, 26 (2006) 10703-10710. Abstract. PDF.
1. Soliton trains and vortex streets as a form of Cerenkov radiation in trapped Bose-Einstein condensates.
R. Carretero-González, P.G. Kevrekidis, D.J. Frantzeskakis. B.A. Malomed, S. Nandi, and A.R. Bishop.
Math. Comput. Simulat., 74 (2007) 361-369. Abstract. PDF.
1. Dynamics and Manipulation of Matter-Wave Solitons in Optical Superlattices.
M.A. Porter, P.G. Kevrekidis, R. Carretero-González, and D.J. Frantzeskakis.
Phys. Lett. A 352 (2006) 210. Abstract. PDF.
1. Vector solitons with an embedded domain wall.
P.G. Kevrekidis, H. Susanto, R. Carretero-González, B.A. Malomed, and D.J. Frantzeskakis.
Phys. Rev. E 72 (2005) 066604. Abstract. PDF.
1. Discrete Solitons and Vortices on Anisotropic Lattices.
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, B.A. Malomed, and A.R. Bishop.
Phys. Rev. E 72 (2005) 046613. Abstract. PDF.
1. Multistable Solitons of the Cubic-Quintic Discrete Nonlinear Schrödinger Equation.
R. Carretero-González, J.D. Talley, C. Chong, and B.A. Malomed.
Physica D 216 (2006) 77-89. Abstract. PDF.
1. Trapped bright solitons in the presence of localized inhomogeneities.
G. Herring, P.G. Kevrekidis, R. Carretero-González, B.A. Malomed, D.J. Frantzeskakis, and A.R. Bishop.
Phys. Lett. A 345 (2005) 144. Abstract. PDF.
1. Three-Dimensional Nonlinear Lattices:
From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes.

R. Carretero-González, P.G. Kevrekidis, B.A. Malomed, and D.J. Frantzeskakis.
Phys. Rev. Lett. 94 (2005) 203901. Abstract. PDF.
1. Multifrequency Synthesis Using two Coupled Nonlinear Oscillator Arrays.
A. Palacios, R. Carretero-González, P. Longhini, N. Renz, V. In, A. Kho, B. Meadows, and J. Neff.
Phys. Rev. E 72 (2005) 026211. Abstract. PDF.
1. Vortices in Bose-Einstein Condensates: Some Recent Developments.
P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, and I.G. Kevrekidis.
Mod. Phys. Lett. B, 18 (2004) 1481-1505. Abstract. PDF.
1. Nonlinear Lattice Dynamics of Bose-Einstein Condensates.
M.A. Porter, R. Carretero-González, 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).
1. Statics, Dynamics and Manipulation of Bright Matter-wave Solitons in Optical Lattices.
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, B.A. Malomed, G. Herring and A.R. Bishop.
Phys. Rev. A, 71 (2005) 023614. Abstract. PDF.
1. Higher-Order Vortices in Nonlinear Dynamical Lattices.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and R. Carretero-González.
Focus on Soliton Research, Editor L.V. Chen, Nova Science Publishers (2006) 139-166. Abstract. PDF.
1. Controlling the motion of dark solitons by means of periodic potentials:
Application to Bose-Einstein condensates in optical lattices.

G. Theocharis, D.J. Frantzeskakis, R. Carretero-González, P.G. Kevrekidis and B.A. Malomed.
Phys. Rev. E 71 (2005) 017602. Abstract. PDF.
1. Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis and R. Carretero-González.
Phys. Rev. Lett. 93 (2004) 080403. Abstract. PDF.
1. Domain Walls of Single-Component Bose-Einstein Condensates in External Potentials.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, A.R. Bishop, H.E. Nistazakis and R. Carretero-González.
Math. Comput. Simulat., 69 (2005) 334-345. Abstract. PDF.
1. A Parrinello-Rahman Approach to Vortex Lattices.
R. Carretero-González, P.G. Kevrekidis, I.G. Kevrekidis, D. Maroudas and D.J. Frantzeskakis.
Phys. Lett. A 341 (2005) 128-134. Abstract. PDF.
1. Precise computations of chemotactic collapse using moving mesh methods.
C.J. Budd, R. Carretero-González and R.D. Russell.
J. Comput. Phys., 202, 2 (2005) 463-487. Abstract. PDF.
1. Families of matter-waves for Two-Component Bose-Einstein Condensates.
P.G. Kevrekidis, G. Theocharis, D.J. Frantzeskakis, B.A. Malomed and R. Carretero-González.
Eur. Phys. J. D: At. Mol. Opt. Phys., 28, 2, (2004) 181-185. Abstract. PDF.
1. Dark soliton dynamics in spatially inhomogeneous media: Application to Bose-Einstein condensates.
G. Theocharis, D.J. Frantzeskakis, R. Carretero-González, P.G. Kevrekidis and B.A. Malomed.
Math. Comput. Simulat. 69 (2005) 537-552. Abstract. PDF.
1. Vortices in a Bose-Einstein condensate confined by an optical lattice.
P.G. Kevrekidis, R. Carretero-González, G. Theocharis, D.J. Frantzeskakis and B.A. Malomed.
J. Phys. B: At. Mol. Phys. 36 (2003) 3467-3476. Abstract. PDF.
1. Stability of dark solitons in a Bose-Einstein condensate trapped in an optical lattice.
P.G. Kevrekidis, R. Carretero-González, G. Theocharis, D.J. Frantzeskakis and B.A. Malomed.
Phys. Rev. A, 68 035602 (2003). Abstract. PDF, [ERRATUM].
1. Variational Mesh Adaptation Methods for Axisymmetrical Problems with Applications to Blowup.
W. Cao, R. Carretero-González, W. Huang and R.D. Russell.
SIAM Journal on Numerical Analysis 41,1 (2003) 235-257. Abstract. PDF.
1. Localized breathing oscillations for Bose-Einstein condensates in periodic traps.
R. Carretero-González and K. Promislow.
Phys. Rev. A 66, 3, 033610 (2002). Abstract. PDF.
1. Stability of attractive Bose-Einstein condensates in a periodic potential.
J.C. Bronski, L.D. Carr, R. Carretero-González, B. Deconinck, J.N. Kutz and K. Promislow.
Phys. Rev. E 64, 5, 056615 (2001). Abstract. PDF.
1. Modelling desert dune fields based on discrete dynamics.
H. Momiji, S.R. Bishop, R. Carretero-González and A. Warren.
Discrete Dynamics in Nature and Society, 7, 1 (2002) 7-17. Abstract. PDF.
1. Simulation of the effect of wind speedup in the formation of transverse dune fields.
H. Momiji, R. Carretero-González, S.R. Bishop and A. Warren.
Earth Surface Processes and Landforms, 25, 8 (2000) 905-918. Abstract. PDF.
1. Quasi-diagonal approach to the estimation of Lyapunov spectra for spatiotemporal systems from multivariate time series.
R. Carretero-González, S. Ørstavik and J. Stark.
Phys. Rev. E 62, 5 (2000) 6429-6439. Abstract. PDF.
1. Thermodynamic limit from small lattices of coupled maps.
R. Carretero-González, S. Ørstavik, J. Huke, D.S. Broomhead and J. Stark.
Phys. Rev. Lett. 83, 18 (1999) 3633-3636. Abstract, PDF.
1. Estimation of intensive quantities in spatio-temporal systems from time-series.
S. Ørstavik, R. Carretero-González and J. Stark.
Physica D 147 (2000) 204-220. Abstract. PDF.
1. Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems.
R. Carretero-González, S. Ørstavik and J. Stark.
Chaos 9, 2 (1999) 466-482 . Abstract, PDF.
1. One-dimensional dynamics for travelling fronts in coupled map lattices.
R. Carretero-González, D.K. Arrowsmith and F. Vivaldi.
Phys. Rev. E 61, 2 (2000) 1329-1336. Abstract, PDF.
1. Low dimensional travelling interfaces in coupled map lattices.
R. Carretero-González.
Int. J. Bifurcation and Chaos 7, 12 (1997) 2745-2754. Abstract, PDF.
1. Mode-Locking in Coupled Map Lattices.
R. Carretero-González, D.K. Arrowsmith and F. Vivaldi.
Physica D 103, 1/4 (1997) 381-403. Abstract, PDF.
1. Regular and Chaotic Behaviour in an Extensible Pendulum.
R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Eur. J. Phys. 15, 3 (1994) 139-148. Abstract, PDF.
1. Evidence of Chaotic Behaviour in Jordan-Brans-Dicke Cosmology.
R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Phys. Lett. A 188, 1 (1994) 48-54. Abstract, PDF.

### Proceedings

1. Optical Manipulation of Matter Waves.
R. Carretero-González, P.G. Kevrekidis, D.J. Frantzeskakis, and B.A. Malomed.
Proc. SPIE Int. Soc. Opt. Eng. 5930 (2005) 59300L. Abstract. PDF.
1. Multifrequency Pattern Generation Using Group-Symmetric Circuits.
J. Neff, V. In, B. Meadows, C. Obra, A. Palacios, and R. Carretero-González,
Submitted to 2006 IEEE International Symposium on Circuits and Systems. Abstract. PDF.
1. The Curvature Criterion and the Dynamics of a Rolling Elastic Cylinder.
M. Arizmendi, R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Proceedings of the "International Symposium on Hamiltonian Systems and Celestial Mechanics", Cocoyoc, Morelos, México, September 13-17, 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) 1-13.
Abstract
1. Nonlinear behaviour in the JBD scalar-tensor theory.
R. Carretero-González, P. Chauvet, H.N. Núñez-Yépez and A.L. Salas-Brito.
Proceedings of the "International Conference on Aspects of General Relativity and Mathematical Physics". Mexico City, June 2-4, 1993.
N.Bretón, R.Capovilla & T.Matos (Eds). CINVESTAV, Mexico City, (1994) 204-209.
Abstract
1. Chaotic behaviour in JBD cosmology.
R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Proceedings of the "8th Latin American Symposium on Relativity and Gravitation", Aguas de Lindoia, Brazil July 25-30, 1993.
Gravitation: The Spacetime Structure.
P.S.Letelier & W.A.Rodrigues (Eds.). Wolrd Scientific, July (1994) 457-461.
Abstract

### Thesis

1. Front propagation and mode-locking in coupled map lattices.
R. Carretero-González.
Ph.D. thesis, Dep. of Mathematical Sciences,
Queen Mary and Westfield College, London, UK, August 1997.

1. The transition to chaos on an extensible pendulum.
R. Carretero-González.
B.Sc. Thesis, Facultad de Ciencias, Universidad Nacional Autónoma de México, México, December 1992. Abstract

### Abstracts

[top]

Dynamics of Few Co-rotating Vortices in Bose-Einstein Condensates.
R. Navarro, R. Carretero-González, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altuntaş, and D.S. Hall.
To appear in Phys. Rev. Lett., 2013. PDF.

We study the dynamics of small vortex clusters with few (2--4) co-rotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counter-intuitive 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 two-vortex system, where a reduction of the phase space of the system provides valuable insight, as well as for the non-integrable three- (or more) vortex case, which additionally admits the possibility of chaotic trajectories.
[top]

Inelastic Collisions of Solitary Waves in Anisotropic Bose-Einstein Condensates:
Sling-Shot Events and Expanding Collision Bubbles.

C. Becker, K. Sengstock, P. Schmelcher, R. Carretero-González, and P.G. Kevrekidis.
To be submitted, 2013. PDF.

We study experimentally and theoretically the dynamics of apparent dark soliton stripes in an elongated Bose-Einstein 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 {\it increasing} time interval between collisions.
[top]

Nonlinear localized modes in two-dimensional electrical lattices.
L.Q. English, F. Palmero, J. Stormes, J. Cuevas, R. Carretero-González, and P.G. Kevrekidis.
Submitted, 2013. 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 self-localized modes were generated experimentally and theoretically in a family of two-dimensional square, as well as honeycomb lattices composed of 6 $\times$ 6 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 PT-symmetric systems with defocusing nonlinearities.
V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González.
Submitted, 2012. PDF.

We examine a prototypical nonlinear Schrödinger model bearing a defocusing nonlinearity and Parity-Time (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 PT-symmetric potential are destabilized via a symmetry breaking (pitchfork) bifurcation. Second, the ground state and the dark soliton die hand-in-hand in a saddle-center bifurcation (a nonlinear analogue of the PT-phase 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 two-soliton structure bearing similar characteristics to the zero-soliton one, and the three-soliton state having the same pitchfork destabilization mechanism and saddle-center collision (in this case with the two-soliton) as the one-dark soliton. All of the above notions are generalized in two-dimensional settings for vortices, where the topological charge enforces the destabilization of a two-vortex state and the collision of a no-vortex state with a two-vortex one, of a one-vortex state with a three-vortex one, and so on. The dynamical manifestation of the instabilities mentioned above is examined through direct numerical simulations.
[top]

Symmetry-breaking Effects for Polariton Condensates in Double-Well Potentials.
A.S. Rodrigues, P.G. Kevrekidis, J. Cuevas, R. Carretero-González, and D.J. Frantzeskakis.
Submitted 2012. PDF.

We study the existence, stability, and dynamics of symmetric and anti-symmetric states of quasi-one-dimensional polariton condensates in double-well 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 double-well 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 sub-critical 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 Two-Dimensional 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. Carretero-González, P.G. Kevrekidis, M.J. Davis, and B.P. Anderson.
Submitted to Phys. Rev. Lett., 2012. PDF. [Suplementary material].

Under suitable forcing a fluid exhibits turbulence, with characteristics strongly affected by the fluid's confining geometry. Here we study two-dimensional quantum turbulence in a highly oblate Bose-Einstein condensate in an annular trap. As a compressible quantum fluid, this system affords a rich phenomenology, allowing coupling between vortex and acoustic energy. Small-scale stirring generates an experimentally observed disordered vortex distribution that evolves into large-scale flow in the form of a persistent current. Numerical simulation of the experiment reveals additional characteristics of two-dimensional quantum turbulence: spontaneous clustering of same-circulation vortices, and an incompressible energy spectrum with k-5/3 dependence for low wavenumbers k and k3 dependence for high k.
[top]

Dark solitons and vortices in PT-symmetric nonlinear media:
from spontaneous symmetry breaking to nonlinear PT phase transitions.

V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González.
Phys. Rev. A 86 (2012) 013808. PDF.

We consider nonlinear analogues of Parity-Time (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 PT-symmetric 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 multi-soliton (multi-vortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear PT-phase transition ---thus termed the nonlinear PT-phase transition. Past this critical point, initialization of, e.g., the former ground state leads to spontaneously emerging solitons and vortices.
[top]

Vortices in Bose-Einstein Condensates: (Super)fluids with a twist.
P.G. Kevrekidis. R. Carretero-González, 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 Bose-Einstein 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 two-body (i.e., two-vortex) system and opens up exciting extensions for N-body generalizations thereof. One-dimensional and three-dimensional analogues of such dynamics, involving dark solitons and vortex rings, respectively, are also briefly touched upon.
[top]

A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and its Application to the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
To appear in J. Comput. Appl. Math., 2012. PDF.

An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-square 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 Two-Step High-Order Compact Scheme for the Laplacian Operator and its Implementation in an Explicit Method for Integrating the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
J. Comput. Appl. Math. 251 (2013) 33-46. PDF.

We describe and test an easy-to-implement two-step high-order 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 `double-differencing' which is shown to be computationally equivalent to standard fourth-order non-compact schemes. Through numerical simulations of the NLSE using fourth-order Runge-Kutta, we confirm that our scheme shows the desired fourth-order accuracy. A computation and storage requirement comparison is made between the 2SHOC scheme and the non-compact 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 Runge-Kutta Finite-Difference Schemes for the Nonlinear Schrödinger Equation.
R.M. Caplan and R. Carretero-González.
App. Num. Math. 71 (2013) 24-40. PDF.

Linearized numerical stability bounds for solving the nonlinear time-dependent Schrödinger equation (NLSE) using explicit finite-differencing are shown. The bounds are computed for the fourth-order Runge-Kutta scheme in time and both second-order and fourth-order central differencing in space. Results are given for Dirichlet, modulus-squared Dirichlet, Laplacian-zero, and periodic boundary conditions for one, two, and three dimensions. Our approach is to use standard Runge-Kutta 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 Bose-Einstein Condensates.
P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, P. Schmelcher, and D.S. Hall.
Phys. Lett. A 375 (2011) 3044-3050. PDF.

We present a systematic theoretical analysis of the motion of a pair of straight counter-rotating vortex lines within a trapped Bose-Einstein 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 guiding-center equilibria about which the vortex lines execute periodic motion; thus, the generic two-vortex motion can be classified as quasi-periodic. 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. Carretero-González, 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, ωd, and the bottom of the linear dispersion curve, ω0. If ωd/2 lies just below ω0, then a single ILM can be generated in a 32-node lattice, whereas when ωd/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 dark-bright solitons in atomic Bose-Einstein condensates.
D. Yan, J.J. Chang, C. Hamner, P.G. Kevrekidis. P. Engels, V. Achilleos, D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
Phys. Rev. A 84 (2011) 053630. PDF.

We present experimental results and a systematic theoretical analysis of dark-bright soliton interactions and multiple-dark-bright soliton complex es in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two-dark-bright 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 dark-bright soliton center and the existence and stability of stationary two-dark-bright soliton states is illustrated (with the bright components being either in- or out-of-phase). 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. Carretero-González, 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 n-peaked 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 n-peaked breathers through modulational instability of the homogeneous steady state. The competition between different discrete breathers seeded by the modulational instability eventually leads to stationary n-peaked solutions whose precise locations is seen to sensitively depend on the initial conditions.
[top]

Guiding-Center Dynamics of Vortex Dipoles in Bose-Einstein Condensates.
S. Middelkamp, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, 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 counter-circulating vortex lines in a superfluid. Although vortex dipoles are endemic in two-dimensional superfluids, the precise details of their dynamics have remained largely unexplored. We present here several striking observations of vortex dipoles in dilute-gas Bose-Einstein condensates, and develop a vortex-particle model that generates vortex line trajectories that are in good agreement with the experimental data. Interestingly, these diverse trajectories exhibit essentially identical quasi-periodic behavior, in which the vortex lines undergo stable epicyclic orbits.
[top]

Nonlinear Excitations, Stability Inversions and Dissipative Dynamics in Quasi-one-dimensional Polariton Condensates.
J. Cuevas, A.S. Rodrigues, R. Carretero-González, 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 quasi-one-dimensional polariton condensate in the presence of non-resonant pumping and nonlinear damping. We find a series of remarkable features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates. For some sizeable parameter ranges, the nodeless ("ground") state becomes unstable towards the formation of stable nonlinear single or multi dark-soliton excitations. It is also observed that for suitable parametric choices, the instability of single dark solitons can nucleate multi-dark-soliton states. Also, for other parametric regions, stable asymmetric sawtooth-like solutions exist. These are shown to emerge through a symmetry-breaking bifurcation from bubble-like 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 multi-soliton dynamical states.
[top]

Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings.
C. Chong, R. Carretero-González, B.A. Malomed, and P.G. Kevrekidis.
Physica D 240 (2011) 1205-1212. PDF.

Solitons of a discrete nonlinear Schrödinger equation which includes the next-nearest-neighbor interactions are studied by means of a variational approximation and numerical computations. A large family of multi-humped solutions, including those with a nontrivial phase structure which are a feature particular to the next-nearest-neighbor 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 matter-wave solitons using the ratchet effect.
M.A. Rietmann, R. Carretero-González, and R. Chacon.
Phys. Rev. A 83 (2011) 053617. PDF.

We demonstrate that directed transport of bright solitons in a quasi-one-dimensional Bose-Einstein 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 matter-wave 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 Bose-Einstein Condensates: a Bifurcation Approach near the Linear Limit.
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
Physica D, 240 (2011) 1449-1459. PDF.

We study the existence and stability properties of clusters of alternating charge vortices in repulsive Bose-Einstein condensates. It is illustrated that such states emerge from cascades of symmetry-breaking bifurcations that can be analytically tracked near the linear limit of the system via weakly nonlinear few-mode 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 Bose-Einstein Condensates.
P.J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, and P.G. Kevrekidis, D.J. Frantzeskakis.
Comm. Pure Appl. Ana. 10 (2011) 1589-1615. PDF.

Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein 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 quasi-periodic (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 Dark-Bright Solitons in Cigar-Shaped Bose-Einstein Condensates.
S. Middelkamp, J.J. Chang, C. Hamner, R. Carretero-González, P.G. Kevrekidis, V. Achilleos, D.J. Frantzeskakis, P. Schmelcher, and P. Engels.
Phys. Lett. A 375 (2011) 642-646. PDF.
Dark-bright (DB) oscillation movies:
[ Movie#1, Fig. 3 ]: Single DB for parameters in Nature Phys. 4, 496 (2008): ND=92,432 and NB=7,973, (fz,fy,fx)=(85,133,5.9) Hz,
[ Movie#2, Fig. 4.a ]: Single DB with bright soliton transverse dynamics: ND=88,181 and NB=1,058, (fz,fy,fx)=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.d ]: Two interacting DBs with out-of-phase (attractive) bright solitons: ND=5,243 and NB=817, (fz,fy,fx)=(133,133,5.9) Hz,
[ Movie#3, Fig. 4.e ]: Two interacting DBs with in-phase (repulsive) bright solitons: ND=5,331 and NB=907, (fz,fy,fx)=(133,133,5.9) Hz.

We explore the stability and dynamics of dark-bright solitons in two-component elongated Bose-Einstein condensates by developing effective 1D vector equations as well as solving the corresponding 3D Gross-Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the dark-bright (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 dark-bright soliton dynamics and collisions in a BEC consisting of two hyperfine states of 87Rb confined in an elongated optical dipole trap are presented.
[top]

Bifurcations, Stability and Dynamics of Multiple Matter-Wave Vortex States.
S. Middelkamp, P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González. 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 quasi-two-dimensional Bose-Einstein condensates. In particular, we illustrate that the multi-vortex 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 few-mode truncation of the system, which clearly showcases the symmetry-breaking nature of the corresponding bifurcation. We complement this small(er) amplitude, few mode bifurcation picture, with a larger amplitude, particle-based 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 non-equilibrium multi-vortex dynamics.
[top]

Controlling the transverse instability of dark solitons and nucleation of vortices by a potential barrier.
Manjun Ma, P.G. Kevrekidis, R. Carretero-González, 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 dark-soliton stripes in two-dimensional (2D) Bose-Einstein condensates (BECs) and self-defocusing bulk optical waveguides by means of quasi-1D 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 dark-soliton 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 Gross-Pitaevskii/nonlinear Schrödinger equation.
[top]

Existence, Stability, and Scattering of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation.
R.M. Caplan, R. Carretero-González. P.G. Kevrekidis, and B.A. Malomed.
Math. Comput. Simulat. 82 (2012) 1150-1171. PDF.

We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex rings) in the two-dimensional (2D) cubic-quintic nonlinear Schrödinger equation. We develop a semi-analytical 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 semi-analytical 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 matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations.
S. Middelkamp, P.G. Kevrekidis, and D.J. Frantzeskakis, R. Carretero-González, and P. Schmelcher.
J. Phys. B 43 (2010) 155303. PDF.

In this work, the spectral properties of a singly-charged vortex in a Bose-Einstein condensate confined in a highly anisotropic (disk-shaped) harmonic trap are investigated. Special emphasis is given on the analysis of the so-called 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 perturbation-induced evolution of the anomalous mode, and being monitored by direct numerical simulations.
[top]

Manipulation of Vortices by Localized Impurities in Bose-Einstein Condensates.
M.C. Davis, R. Carretero-González, 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 Bose-Einstein 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 single-charge 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 Two-component Bose-Einstein Condensates.
R. Navarro, R. Carretero-González, 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 Bose-Einstein 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 one-dimensional, two-atomic 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 non-topological, phase-separated states of increasing complexity emerge out of a symmetric state, as the interspecies coupling is increased. The symmetry-breaking dynamical evolution of some of these states is numerically monitored and the associated asymmetric states are also explored.
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Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation.
R.M. Caplan, Q.E. Hoq, R. Carretero-González, and P.G. Kevrekidis.
Optics. Comm. 282 (2009) 1399-1405. PDF.

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional 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 quasi-one-dimensional 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.
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Spinor Bose-Einstein condensate past an obstacle.
S. S. Rodrigues, P.G. Kevrekidis, R. Carretero-González, 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) Bose-Einstein condensate in the presence of an obstacle. We consider both cases of ferromagnetic and polar spin-dependent 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 vortex-antivortex pairs, as well as vortex rings in one- and higher-dimensional settings respectively, when a localized defect (e.g., a blue-detuned laser beam) is dragged through the spinor condensate at a speed larger than the second critical one.
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Dissipative Solitary Waves in Periodic Granular Crystals.
R. Carretero-González, 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 one-dimensional 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 quantitatively-accurate 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 material-dependent prefactor.
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Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice.
R. Chacon, D. Bote, and R. Carretero-González.
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 Bose-Einstein 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.
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A Map Approach to Stationary Solutions of the Discrete Nonlinear Schrödinger Equation.
R. Carretero-González.
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 well-established map approach for obtaining stationary solutions to the one-dimensional (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 two-dimensional (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.
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Structure and stability of two-dimensional Bose-Einstein condensates under both harmonic and lattice confinement.
K.J.H. Law, P.G. Kevrekidis, B.P. Anderson, R. Carretero-González, and D.J. Frantzeskakis.
J. Phys. B, 41 (2008) 195303. PDF.

In this work, we study pancake-shaped Bose-Einstein 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 two-dimensional linear problem through multiple-scale 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. Two-parameter 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.
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Surface Solitons in Three Dimensions.
Q.E. Hoq, R. Carretero-González, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, Yu.V. Bludov, and V.V. Konotop.
To appear in Phys. Rev. E, 2008. PDF.

We study localized modes on the surface of a three-dimensional 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 three-site "horseshoe"-shaped structure, which is always unstable in the bulk, while at the surface it is stable near the anti-continuum 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 five-site horseshoes and pyramids, are also considered.
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Multistable Solitons in Higher-Dimensional Cubic-Quintic Nonlinear Schrödinger Lattices.
C. Chong, R. Carretero-González, B.A. Malomed, and P.G. Kevrekidis.
Physica D, 238 (2009) 126-136. PDF.

We study the existence, stability, and mobility of fundamental discrete solitons in two- and three-dimensional nonlinear Schrödinger lattices with a combination of cubic self-focusing and quintic self-defocusing onsite nonlinearities. Several species of stationary solutions are constructed, and bifurcations linking their families are investigated using parameter continuation starting from the anti-continuum limit, and also with the help of a variational approximation. In particular, a new species of hybrid solitons, intermediate between the site- and bond-centered 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 multi-dimensional discrete solitons that can be set in motion by lending them kinetic energy exceeding the appropriately crafted Peierls-Nabarro barrier; however, they eventually come to a halt, due to radiation loss.
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Nonlinear dynamics of Bose-condensed gases by means of a q-Gaussian variational approach.
A.I. Nicolin and R. Carretero-González.
Physica A 387 (2008) 6032. Abstract. PDF. PDF.

We propose a versatile variational method to investigate the spatio-temporal dynamics of one-dimensional magnetically-trapped Bose-condensed gases. To this end we employ a q-Gaussian trial wave-function that interpolates between the low- and the high-density limit of the ground state of a Bose-condensed gas. Our main result consists of reducing the Gross-Pitaevskii 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 wave-function and a separate algebraic equation yielding the generalized width. Our equations recover those of the usual Gaussian variational approach (in the low-density regime), and the hydrodynamic equations that describe the high-density regime. Finally, we show a detailed comparison between the numerical results of our equations and those of the original Gross-Pitaevskii equation.
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Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity.
F. Palmero, R. Carretero-González, 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 one-dimensional 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 on-site mode centered at the defect in the repulsive case; b) the disappearance of localized modes in the vicinity of the defect due to saddle-node 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.
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Dynamics of Vortex Formation in Merging Bose-Einstein Condensate Fragments.
R. Carretero-González, 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 Bose-Einstein 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 three-dimensional 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.
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Resonant energy transfer in Bose-Einstein condensates.
A.I. Nicolin, M.H. Jensen, J.W. Thomsen, and R. Carretero-González.
Physica D, 237 (2008) 2476-2481. PDF.

We consider the dynamics of a dilute, magnetically-trapped one-dimensional Bose-Einstein 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 time-dependent 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 Gross-Pitaevskii equation.
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Nonlinear Waves in Bose-Einstein Condensates:
Physical Relevance and Mathematical Techniques.

R. Carretero-González, D.J. Frantzeskakis, and P.G. Kevrekidis.
Nonlinearity 21 (2008) R139-R202. 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 Bose-Einstein 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.
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Radially Symmetric Nonlinear States of Harmonically Trapped Bose-Einstein Condensates.
G. Herring, L.D. Carr, R. Carretero-González, 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 nr=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 nr=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.
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A.I. Nicolin, R. Carretero-González, and P.G. Kevrekidis.
Phys. Rev. A 76 (2007) 063609. PDF.

Motivated by recent experiments on Faraday waves in Bose-Einstein condensates we investigate both analytically and numerically the dynamics of cigar-shaped Bose-condensed 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 Mathieu-type analysis of the non-polynomial 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 one-dimensional non-polynomial Schrödinger equation and of the fully three-dimensional Gross-Pitaevskii equation.
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Extended Nonlinear Waves in Multidimensional Dynamical Lattices.
Q.E. Hoq, J. Gagnon, P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and R. Carretero-González.
Math. Comput. Simulat., 80 (2009) 721-731. PDF.

We explore spatially extended dynamical states in the discrete nonlinear Schrödinger lattice in two- and three- dimensions, starting from the anti-continuum limit. We first consider the "core" of the relevant states (either a two-dimensional "tile" or a three-dimensional "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".
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Polarized States and Domain Walls in Spinor Bose-Einstein Condensates.
H.E. Nistazakis, D.J. Frantzeskakis. P.G. Kevrekidis, B.A. Malomed, and R. Carretero-González, and A.R. Bishop.
Phys. Rev. A 76 (2007) 063603. PDF.

We study spin-polarized states and their stability in anti-ferromagnetic states of spinor (F=1) quasi-one-dimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we find various types of polarized states, including: patterns of the Thomas-Fermi type; structures with a pulse-shape in one component inducing a hole in the other components; states with holes in all three components; and domain walls. A Bogoliubov-de 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.
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Bright-Dark Soliton Complexes in Spinor Bose-Einstein Condensates.
H.E. Nistazakis, D.J. Frantzeskakis. P.G. Kevrekidis, B.A. Malomed, and R. Carretero-González.
Phys. Rev. A 77 (2008) 033612. PDF.

We present novel solutions for bright-dark vector solitons in quasi-one-dimensional spinor (F=1) Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding system of three coupled Gross-Pitaevskii equations (GPEs) to a completely integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the mF=+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 quasi-elastic 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.
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Symmetry breaking in linearly coupled dynamical lattices.
G. Herring, P.G. Kevrekidis. B.A. Malomed, R. Carretero-González, and D.J. Frantzeskakis.
Phys. Rev. E 76 (2007) 066606. PDF.

We examine one- and two-dimensional (1D and 2D) models of linearly coupled lattices of the discrete-nonlinear-Schrödinger type. Analyzing ground states of the systems with equal powers in the two components, we find a symmetry-breaking phenomenon beyond a critical value of the squared L2-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).
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Non-Equilibrium Dynamics and Superfluid Ring Excitations in Binary Bose-Einstein Condensates.
K.M. Mertes, J. Merrill, R. Carretero-González, 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 Bose-Einstein condensates in the hyperfine states |F=1,mf=-1>=|1> and |F=2,mf=+1>=|2> in  87Rb and observe striking new non-equilibrium component separation dynamics in the form of oscillating ring-like 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 multi-component mean-field 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).
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Čerenkov-like radiation in a binary superfluid flow past an obstacle.
H. Susanto, P.G. Kevrekidis. R. Carretero-González, 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 Bose-Einstein 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 dark-antidark soliton in 1D; the 2D analog of such a structure, a vortex-lump, is also observed.
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Vortex Structures Formed by the Interference of Sliced Condensates.
R. Carretero-González, 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 higher-dimensional Bose-Einstein condensates (BECs). This study is motivated by earlier theoretical results pertaining to the formation of dark solitons by interfering quasi one-dimensional BECs, as well as recent experiments demonstrating the formation of vortices by interfering higher-dimensional BECs. Here, we demonstrate the genericity of the relevant scenario, but also highlight a number of additional possibilities emerging in higher-dimensional settings. A relevant example is, e.g., the formation of a "cage" of vortex rings surrounding the three-dimensional 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.
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Mode locking of a driven Bose-Einstein condensate.
A.I. Nicolin, M.H. Jensen, and R. Carretero-González.
Phys. Rev. E 75 (2007) 036208. PDF.

We consider the dynamics of a driven Bose-Einstein condensate with positive scattering length. Employing an accustomed variational treatment we show that when the scattering length is time-modulated 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 auto-resonance, 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 finite-depth energy-minimum of an effective potential.
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Rotating matter waves in Bose-Einstein condensates.
T. Kapitula, P.G. Kevrekidis, and R. Carretero-González.
Physica D, 233 (2007) 112-137. PDF.

In this paper we consider analytically and numerically the dynamics of waves in two-dimensional, magnetically trapped Bose-Einstein condensates in the weak interaction limit. In particular, we consider the existence and stability of azimuthally modulated structures such as rings, multi-poles, soliton necklaces, and vortex necklaces. We show how such structures can be constructed from the linear limit through Lyapunov-Schmidt 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.
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Discrete surface solitons in two dimensions.
H. Susanto, P.G. Kevrekidis, B.A. Malomed, R. Carretero-González, 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.
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Mobility of Discrete Solitons in Quadratic Nonlinear Media.
H. Susanto, P.G. Kevrekidis, R. Carretero-González, 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 second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro 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 one-dimensional (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.
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Skyrmion-like states in two- and three-dimensional dynamical lattices.
P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, B.A. Malomed, and F.K. Diakonos.
Phys. Rev. E 75 (2007) 026603. PDF.

We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the 2D case, an analog of the baby-Skyrmion 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 quasi-Skyrmion on the cubic lattice is a toroidal structure, composed of a nearly planar complex-field discrete vortex and a 3D real-field 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.
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Multipole-mode solitons in Bessel optical lattices.
Y.V. Kartashov, R. Carretero-González, B.A. Malomed, V.A. Vysloukh, and Ll. Torner.
Optics Express 13, 26 (2006) 10703-10710. PDF.

We study basic properties of multipole-mode 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 high-power limit, the multipole-mode solitons feature a multi-ring structure.
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Soliton trains and vortex streets as a form of Cerenkov radiation in trapped Bose-Einstein condensates.
R. Carretero-González, P.G. Kevrekidis, D.J. Frantzeskakis. B.A. Malomed, S. Nandi, and A.R. Bishop.
Math. Comput. Simulat., 74 (2007) 361-369. PDF.

We numerically study the nucleation of gray solitons and vortex-antivortex pairs created by a moving impurity in, respectively, 1D and 2D Bose-Einstein condensates (BECs) confined by a parabolic potential. The simulations emulate the motion of a localized laser-beam 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 square-root 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 vortex-antivortex 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.
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Dynamics and Manipulation of Matter-Wave Solitons in Optical Superlattices.
M.A. Porter, P.G. Kevrekidis, R. Carretero-González, and D.J. Frantzeskakis.
Phys. Lett. A 352 (2006) 210. PDF.

We consider matter-wave 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 (time-dependent) "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".
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Vector solitons with an embedded domain wall.
P.G. Kevrekidis, H. Susanto, R. Carretero-González, 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 Bose-Einstein condensates (BECs) with inter-species 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 intra-species interaction, a bright-dark 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.
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Discrete Solitons and Vortices on Anisotropic Lattices.
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, 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 two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation, which predicts that broad quasi-continuum 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 on-site-centered 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 so-called "super-symmetric" intersite-centered 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 Lyapunov-Schmidt 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.
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Multistable Solitons of the Cubic-Quintic Discrete Nonlinear Schrödinger Equation.
R. Carretero-González, J.D. Talley, C. Chong, and B.A. Malomed.
Physica D 216 (2006) 77-89. PDF.

We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schrödinger (DNLS) equation with a combination of self-focusing cubic and defocusing quintic on-site 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 saddle-node bifurcation. Salient properties of the model, which distinguish it from the well-known 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.
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Trapped bright solitons in the presence of localized inhomogeneities.
G. Herring, P.G. Kevrekidis, R. Carretero-González, 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 Bose-Einstein 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 saddle-node 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.
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Three-Dimensional Nonlinear Lattices:
From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes.

R. Carretero-González, 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 Bose-Einstein 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 quasi-planar vortices) and "diamonds" (formed by two orthogonal vortices).
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Multifrequency Synthesis Using two Coupled Nonlinear Oscillator Arrays.
A. Palacios, R. Carretero-González, 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 symmetry-breaking bifurcations for generating a spatio-temporal 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 master-slave interaction, a synchronization effect, or that of sub-harmonic and ultra-harmonic 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.
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Vortices in Bose-Einstein Condensates: Some Recent Developments.
P.G. Kevrekidis, R. Carretero-González, D.J. Frantzeskakis, and I.G. Kevrekidis.
Mod. Phys. Lett. B, 18 (2004) 1481-1505. PDF.

In this brief review, we summarize a number of recent developments in the study of vortices in Bose-Einstein 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 two-component condensates. Last but not least, we discuss clusters of vortices, the so-called vortex lattices and analyze some of their intriguing dynamical features. A number of interesting future directions are highlighted.
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Nonlinear Lattice Dynamics of Bose-Einstein Condensates.
M.A. Porter, R. Carretero-González, 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 Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic 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, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems---including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular, due to their widely ranging and easily manipulated dynamical apparatuses---with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials---provide 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.
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Statics, Dynamics and Manipulation of Bright Matter-wave Solitons in Optical Lattices.
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González, 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 Bose-Einstein 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 Lyapunov-Schmidt 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.
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Higher-Order Vortices in Nonlinear Dynamical Lattices.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, and R. Carretero-González.
Focus on Soliton Research, Editor L.V. Chen, Nova Science Publishers (2006) 139-166. PDF.

In this paper, we investigate localized discrete states with a non-zero topological charge (discrete vortices) in the prototypical model of dynamical-lattice systems, based on the two- and three-dimensional (2D and 3D) discrete nonlinear Schrödinger (DNLS) equation, with both attractive and repulsive on-site 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). Quasi-vortices, 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 gap-soliton 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 two-component system, stable compound vortices of the type (S1,S2) =(1,±1) 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 two-component complex with mutually orthogonal vortices in the components. Applications of the results to nonlinear optics optics and Bose-Einstein condensates are briefly discussed.
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Controlling the motion of dark solitons by means of periodic potentials:
Application to Bose-Einstein condensates in optical lattices.

G. Theocharis, D.J. Frantzeskakis, R. Carretero-González, 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 cigar-shaped Bose-Einstein 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.
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Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis and R. Carretero-González.
Phys. Rev. Lett. 93 (2004) 080403. PDF.

In a benchmark dynamical-lattice 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 two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices, and in photonic crystals built of microresonators.
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Domain Walls of Single-Component Bose-Einstein Condensates in External Potentials.
P.G. Kevrekidis, B.A. Malomed, D.J. Frantzeskakis, A.R. Bishop, H.E. Nistazakis and R. Carretero-González.
Math. Comput. Simulat., 69 (2005) 334-345. 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 optical-lattice 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.
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A Parrinello-Rahman Approach to Vortex Lattices.
R. Carretero-González, P.G. Kevrekidis, I.G. Kevrekidis, D. Maroudas and D.J. Frantzeskakis.
Phys. Lett. A 341 (2005) 128-134. PDF.

We present a framework for studying vortex lattice patterns and their structural transitions, using the Parrinello-Rahman (PR) method for Molecular-Dynamics (MD) simulations. Assuming an interaction between vortices derived from a Ginzburg-Landau field-theoretic context, we extract the ground-state of a "vortex gas'' using the PR-MD 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..
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Precise computations of chemotactic collapse using moving mesh methods.
C.J. Budd, R. Carretero-González and R.D. Russell.
J. Comput. Phys., 202, 2 (2005) 463-487. PDF.

We study the application of dynamic (scale-invariant) 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), 583-623).
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Families of matter-waves for Two-Component Bose-Einstein Condensates.
P.G. Kevrekidis, G. Theocharis, D.J. Frantzeskakis, B.A. Malomed and R. Carretero-González.
Eur. Phys. J. D: At. Mol. Opt. Phys., 28, 2, (2004) 181-185. PDF.

We produce several families of solutions for two-component nonlinear Schrödinger/Gross-Pitaevskii 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 Bose-Einstein 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).
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Dark soliton dynamics in spatially inhomogeneous media: Application to Bose-Einstein condensates.
G. Theocharis, D.J. Frantzeskakis, P.G. Kevrekidis, R. Carretero-González and B.A. Malomed.
Math. Comput. Simulat. 69 (2005) 537-552. PDF.

We study the dynamics of dark solitons in spatially inhomogeneous media with app lications to cigar-shaped Bose-Einstein 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.
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Vortices in a Bose-Einstein condensate confined by an optical lattice.

P.G. Kevrekidis, R. Carretero-González, G. Theocharis, D.J. Frantzeskakis and B.A. Malomed.
J. Phys. B: At. Mol. Phys. 36 (2003) 3467-3476. PDF.

We investigate dynamics of vortices in repulsive Bose-Einstein 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 anti-phase solitons.
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Stability of dark solitons in a Bose-Einstein condensate confined in an optical lattice.

P.G. Kevrekidis, R. Carretero-González, 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 one-dimensional Bose-Einstein condensate in the presence of the magnetic parabolic trap and an optical lattice (OL). The analysis is based on both the full Gross-Pitaevskii equation and its tight-binding 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 quasi-periodic oscillations of the DS about the minimum of the parabolic trap.
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Variational Mesh Adaptation Methods for Axisymmetrical Problems with Applications to Blowup.
W. Cao, R. Carretero-González, W. Huang and R.D. Russell.
SIAM Journal on Numerical Analysis 41,1 (2003) 235-257. 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 , where and are the radial and angular unit vectors.

It is shown that when higher mesh concentration at the origin is desired, a choice of and satisfying will make the mesh denser at than in the surrounding area whether or not has a maximum value at . The purpose can also be served by choosing to have a local maximum at when a Winslow-type monitor function with is employed. On the other hand, it is shown that the traditional functional provides little control over mesh concentration around a ring by choosing and .

In contrast, numerical results show that the new functional provides better control of the mesh concentration through the monitor function. Two-dimensional numerical results are presented to support the analysis.
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Localized breathing oscillations for Bose-Einstein condensates in periodic traps.
R. Carretero-González and K. Promislow.
Phys. Rev. A 66, 3, 033610 (2002). PDF.

We demonstrate the existence of localized oscillatory breathers for quasi-one-dimensional Bose-Einstein condensates confined in periodic potentials. The breathing behavior corresponds to position-oscillations of individual condensates about the minima of the potential lattice. Localized oscillations are identified with homoclinic tangles of a reduced two-dimensional 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 Gross-Pitaevskii equation in one dimension.
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Stability of attractive Bose-Einstein condensates in a periodic potential.

J.C. Bronski, L.D. Carr, R. Carretero-González, 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 quasi-one-dimensional repulsive dilute gas Bose-Einstein 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 trivial-phase 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 Thomas-Fermi limit.
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Modelling desert dune fields based on discrete dynamics.

H. Momiji, S.R. Bishop, R. Carretero-González and A. Warren.
Discrete Dynamics in Nature and Society, 7, 1 (2002) 7-17. 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.
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Simulation of the effect of wind speedup in the formation of transverse dune fields.

H. Momiji, R. Carretero-González, S.R. Bishop and A. Warren.
Earth Surface Processes and Landforms, 25, 8 (2000) 905-918. PDF.

A computer simulation model for transverse-dune-field dynamics, corresponding to a uni-directional wind regime, is developed. In a previous formulation, two distinct problems were found regarding the cross-sectional 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 non-linear shear-velocity-increase 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 non-linear 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 inter-dune morphology more precise dynamics in the lee of the dune must be incorporated.
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Quasi-diagonal approach to the estimation of Lyapunov spectra for spatiotemporal systems from multivariate time series.

R. Carretero-González, S. Ørstavik and J. Stark.
Phys. Rev. E 62, 5 (2000) 6429-6439. PDF.

We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series 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 high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatio-temporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying.
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Thermodynamic limit from small lattices of coupled maps.

R. Carretero-González, S. Ørstavik, J. Huke, D.S. Broomhead and J. Stark.
Phys. Rev. Lett. 83, 18 (1999) 3633-3636. 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 two-point correlation with increasing truncated lattice size. This suggests that spatio-temporal 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.
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Estimation of intensive quantities in spatio-temporal systems from time-series.

S. Ørstavik, R. Carretero-González and J. Stark.
Physica D 147 (2000) 204-220. PDF.

We study multi-variate time-series generated by coupled map lattices exhibiting spatio-temporal 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 spatio-temporally chaotic regime and paying careful attention to errors introduced by sub-system 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.
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Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems.

R. Carretero-González, S. Ørstavik and J. Stark.
Chaos 9, 2 (1999) 466-482 . 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 spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system 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 one-dimensional 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 sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system 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 Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.
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One-dimensional dynamics for travelling fronts in coupled map lattices.

R. Carretero-González, D.K. Arrowsmith and F. Vivaldi.
Phys. Rev. E 61, 2 (2000) 1329-1336. 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 infinitely-dimensional dynamics to a one-dimensional circle homeomorphism, whose rotation number gives the propagation velocity. The mode-locking 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 mode-locking tends to disappear as we approach the continuum limit of an infinite density of sites.
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Low dimensional travelling interfaces in coupled map lattices.

R. Carretero-González.
Int. J. Bifurcation and Chaos 7, 12 (1997) 2745-2754. PDF.

We study the dynamics of the travelling interface arising from a bistable piece-wise linear one-way 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 mode-locking with respect to the system parameters. We analytically compute the Arnold's tongues where particular spatio-temporal periodic orbits exist. The boundaries of the mode-locked regions correspond to border-collision bifurcations of the toral map. By varying the system parameters it is possible to increase the number of interfacial sites corresponding to a border-collision bifurcation of the interfacial attracting cycle. We finally give some generalizations towards smooth coupled map lattices whose interface dynamics is typically infinite dimensional
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Mode-Locking in Coupled Map Lattices.

R. Carretero-González, D.K. Arrowsmith and F. Vivaldi.
Physica D 103, 1/4 (1997) 381-403. PDF.

We study propagation of pulses along one-way coupled map lattices, which originate from the transition between two superstable states of the local map. The velocity of the pulses exhibits a staircase-like behaviour as the coupling parameter is varied. For a piece-wise 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 mode-locking for circle maps. We provide evidence that mode-locking is also present for a broader range of maps and couplings.
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Regular and Chaotic Behaviour in an Extensible Pendulum.

R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Eur. J. Phys. 15, 3 (1994) 139-148. 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 low-energy regular motion and the coexistence of stochastic and regular motion at intermediate energies. We employ other diagnostic techniques for checking our conclusions.
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Evidence of Chaotic Behaviour in Jordan-Brans-Dicke Cosmology.

R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Phys. Lett. A 188, 1 (1994) 48-54. Abstract,

We study numerically the properties of solutions of spatially homogeneous Bianchi-type IX cosmological models in the Jordan-Brans-Dicke 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 Jordan-Brans-Dicke cosmology. The range of values of the Jordan-Brans-Dicke coupling parameter considered is (-500,-1).
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Optical Manipulation of Matter Waves.
R. Carretero-González, 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 Bose-Einstein condensation (BEC) has precipitated an intense effort to understand and control interactions of nonlinear matter-waves. Key ingredients in manipulations of matter-waves 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 time-dependent 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.
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Multifrequency Pattern Generation Using Group-Symmetric Circuits.
J. Neff, V. In, B. Meadows, C. Obra, A. Palacios, and R. Carretero-González,
Submitted to 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, group-theoretic arguments are used to dictate the particular coupling topology between the unit bi-stable 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.
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The Curvature Criterion and the Dynamics of a Rolling Elastic Cylinder.

M. Arizmendi, R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
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) 1-13.

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 order-chaos-order 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.
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Nonlinear behaviour in the JBD scalar-tensor theory.
R. Carretero-González, P. Chauvet, H.N. Núñez-Yépez and A.L. Salas-Brito.
Proceedings of the "Int. Conference on Aspects of General Relativity and Mathematical Physics". Mexico City, June 2-4, 1993.
N.Bretón, R.Capovilla & T.Matos (Eds). CINVESTAV, Mexico City, (1994) 204-209.

We apply techniques of nonlinear dynamics to a cosmological problem in the Jordan Brans-Dicke 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.
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Chaotic behaviour in JBD cosmology.

R. Carretero-González, H.N. Núñez-Yépez and A.L. Salas-Brito.
Proceedings of the "8th Latin American Symposium on Relativity and Gravitation", Aguas de Lindoia, Brazil July 25-30, 1993.
Gravitation: The Spacetime Structure.
P.S.Letelier & W.A.Rodrigues (Eds.). Wolrd Scientific, July (1994) 457-461.

Solutions for the spatially homogeneous Bianchi-type IX cosmological model are studied in the Jordan Brans-Dicke 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 Bianchi-IX model in general relativity. It seems as if the main contribution to stochasticity came from the oscillating approach to the singularity.
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Front propagation and mode-locking in coupled map lattices.

R. Carretero-González.
Ph.D. thesis, Dep. of Mathematical Sciences,
Queen Mary and Westfield College, London, UK, August 1997.

We study the propagation of coherent signals through bistable one-way 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 non-decreasing 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 well-defined velocity that depends on the coupling parameter $\e$. For some local maps the velocity is shown to have $\e$-intervals where it is mode-locked to a rational value.

In order to understand the mode-locking phenomenon we introduce a continuous piece-wise linear local map. We show how the dynamics of the whole lattice (infinite system) may be reduced to a one-dimensional auxiliary map. The auxiliary map is a circle-like map whose rotation number corresponds to the velocity of the travelling interface. We introduce symbolic dynamics to fully understand the mode-locking 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 mode-locked to rational plateaus and may be fully described via Farey sequences and modular transformations.

Finally we give some numerical examples depicting mode-locking 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 mode-locking 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.
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The transition to chaos on an extensible pendulum.

R. Carretero-González.
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).
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Scitation Index

Math Reviews of some of my papers