Adiabatic N-soliton interactions of Bose-Einstein condensates in external potentials

A perturbed version of the complex Toda chain (CTC) has been employed to describe adiabatic interactions within an N-soliton train of the nonlinear Schrödinger equation. Perturbations induced by weak quadratic and periodic external potentials are studied by both analytical and numerical means. It is found that the perturbed CTC adequately models the N-soliton train dynamics for both types of potentials. As an application of the developed theory, we consider the dynamics of a train of matter-wave solitons confined to a parabolic trap and optical lattice, as well as tilted periodic potentials. In the last case, we demonstrate that there exist critical values of the strength of the linear potential for which one or more localized states can be extracted from a soliton train. An analytical expression for these critical strengths for expulsion is also derived.