Construction of a multi-soliton-like solutions for non-integrable Schr\"odinger equations with non-trivial far field

TL;DR

This paper proves the asymptotic stability of dark soliton chains and constructs multi-soliton-like solutions for non-integrable 1D Schrödinger equations, revealing their large-time behavior resembles decoupled solitons.

Contribution

It introduces a method to establish stability and construct multi-soliton solutions for non-integrable Schrödinger equations with non-zero boundary conditions.

Findings

Proved asymptotic stability of dark soliton chains.

Constructed exact multi-soliton-like solutions.

Showed large-time dynamics resemble decoupled solitons.

Abstract

This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schr\"odinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.