Asymptotic stability of soliton for discrete nonlinear Schr\"odinger equation on one-dimensional lattice

TL;DR

This paper presents a concise proof demonstrating the asymptotic stability of solitons in the discrete nonlinear Schrödinger equation on a one-dimensional lattice, especially near the anti-continuous limit.

Contribution

It introduces a novel approach by reducing the analysis of a non-symmetric linearized operator to a self-adjoint operator similar to the free discrete Laplacian.

Findings

Proves asymptotic stability of solitons near anti-continuous limit

Simplifies the analysis of linearized operators in discrete NLS

Provides a new method for studying stability in lattice models

Abstract

In this paper we give a simple and short proof of asymptotic stability of soliton for discrete nonlinear Schr\"odinger equation near anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually non-symmetric, can be reduced to a study of simple self-adjoint operator almost like the free discrete Laplacian restricted on odd functions.