Asymptotic stability of $N$-solitons in the cubic NLS equation

TL;DR

This paper proves the asymptotic stability of N-solitons in the cubic focusing NLS equation by analyzing the decay of residuals using inverse scattering and the ar methods.

Contribution

It introduces a novel application of inverse scattering and ar techniques to establish the decay and stability of N-solitons for the cubic NLS.

Findings

Decay of the residual term in L^{ty} norm over time

Asymptotic stability of N-solitons demonstrated

Methodology applicable to similar nonlinear wave equations

Abstract

In this article we consider the Cauchy problem for the cubic focusing nonlinear Schr\"o\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.