On asymptotic stability of N-solitons of the defocusing nonlinear Schrodinger equation

TL;DR

This paper analyzes the asymptotic stability of N-solitons in the defocusing nonlinear Schrödinger equation, deriving precise long-time behavior and stability results for solutions close to solitons using advanced analytical methods.

Contribution

It introduces a novel application of the DBAR steepest descent method to establish asymptotic stability of N-solitons in the defocusing NLS.

Findings

Derived leading order approximation for GP solutions in the solitonic region

Established decay bounds for errors as time tends to infinity

Proved asymptotic stability for initial data near N-dark solitons

Abstract

We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic region of space time $|x| < 2t$ for large times and provide bounds for the error which decay as $t \to \infty$ for a general class of initial data whose difference from the non-vanishing background possess's a fixed number of finite moments and derivatives. Using properties of the scattering map for (GP) we derive as a corollary an asymptotic stability result for initial data which are sufficiently close to the N-dark soliton solutions of (GP).