Soliton trains and their interactions in nonlinear media

The Nonlinear Shrödinger equation (NLS):

$$i u_t + {1 \over 2} u_{xx} + \vert u\vert^2 u = 0,$$ (1)

accepts exact multi-soliton solutions [1]. However, these formulae quickly become prohibitive even for a handful of interacting solitons. Thus an alternative approach is needed in order to follow the interaction of large numbers of solitons. The idea is to approximate every soliton of the multi-soliton solution by a one-soliton solution: $u_k(x,t)= \nu_k(t) {\rm sech}(\nu_k(t)(x -\xi_k(t))) e^{i[\mu_k(t)(x -\xi_k(t))+\delta_k(t)]},$ with time dependent parameters: position $\xi_k(t)$, width $\nu_k(t)$, velocity $\mu_k(t)$ and phase $\delta_k(t)$. With this ansatz it is possible to apply perturbation techniques [2] and/or variational methods [3] to determine the evolution of the soliton parameters. The originally infinite-dimensional system (a PDE) is then reduced to a lattice of coupled ODEs, or lattice differential equation (LDE), on the soliton parameters:

$$(\dot\xi_k(t),\dot\nu_k(t),\dot\mu_k(t),\dot\delta_k(t)) = F (\xi_k(t),\nu_k(t),\mu_k(t),\delta_k(t)).$$ (2)

Using this reduction it is possible to apply dynamical systems ideas to study the possible interactions in the soliton chain.

Applications to fiber optics communication: Hyper-solitons [t.4]
Using the LDE reduction it is possible to obtain novel soliton dynamics describing the propagation of a compressive wave in the soliton positions through the lattice. This compressive wave corresponds to a soliton of the LDE that in turn translates as a soliton of solitons, a Hyper-soliton, of the original PDE (1).

Applications to Bose-Einstein condensates: Localized Breathers [s.2,t.4,w.4]
By adding a periodic potential $V(x)$ to the NLS

$$i u_t + {1 \over 2} u_{xx} + \vert u\vert^2 u = V(x) u,$$ (3)

one may describe the behaviour of interacting aggregate of coherent matter --Bose-Einstein Condensates (BECs) [4]-- under certain physical conditions. After reducing the governing PDE we obtain a new LDE that describes the interactions between neighboring BECs pinned on a regular lattice. We show how the inclusion of the periodic potential allows for the existence of time-periodic, exponentially localized vibrations: breathers. We assume the parameters of each soliton oscillating with a common frequency $\omega$ but different amplitudes $A_k$. We further reduce the dynamics and find a two-dimensional map relating $A_{k+1}$ to $A_{k}$ and $A_{k-1}$. The exponentially localized vibration correspond to homoclinic connections or tangles of the stable and unstable manifolds of the origin [5].

Future directions: Extended vibrational solutions of the BECs and their stability [w.4]
Using the above approach and looking for more general kinds of connections or tangles (homoclinic or heteroclinic) emanating from points away from the origin, it should be possible to obtain extended (i.e. no longer localized) vibrational modes of the soliton parameters. This technique would allow us to construct a much richer plethora of solutions. In the same spirit, I would like to investigate the possibility of deducing the stability of the above solutions using the dynamical reduction approach.