Abstract: In this paper, a perturbed complex Toda chain has been employed to describe the adiabatic interactions in an N-soliton train of the Gross-Pitaevskii equation. Perturbations induced by weak quadratic and periodic external potentials are analytically and numerically studied. It is found that the perturbed complex Toda chain 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 in a quadratic 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 or periodic potential for which one or more localized states can be extracted from a soliton train. In addition, some interesting results in the experiments and applications of the Bose-Einstein condensates are also obtained.

Keywords: 
• Bose-Einstein condensates /
• Gross-Pitaevskii equation /
• matter-wave solitons /
• interactions