Hamilton theory of NLS equation soliton motion

TL;DR

This paper develops a Hamiltonian framework for describing soliton motion in non-uniform, time-dependent backgrounds within integrable wave equations, specifically applied to the defocusing nonlinear Schrödinger equation.

Contribution

It introduces a novel method based on Stokes' argumentation to derive Hamilton equations for soliton dynamics in variable backgrounds, extending existing approaches.

Findings

Derived Hamilton equations for soliton motion in variable backgrounds.

Formulated conditions for external potential influence on background evolution.

Obtained Newtonian equations governing soliton dynamics.

Abstract

We suggest the method of derivation of Hamilton equations which describe the motion of solitons along non-uniform and time dependent large-scale background in case of wave dynamics described by the completely integrable equations in the Ablowitz-Kaup-Newell-Segur scheme. The method is based on development of old Stokes' argumentation which allows one to continue analytically some relationships derived for linear waves to the soliton region. It is presented here for a particular case of the defocusing nonlinear Schr\"{o}dinger equation. We formulate the condition when the external potential should only be taken into account for the background evolution, and in this case we obtain the Newton equation for the soliton dynamics.