Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing NLS equation with a delta potential and even initial data, using Riemann-Hilbert problem techniques to study soliton stability.

Contribution

It extends previous methods by applying the nonlinear steepest-descent approach to the NLS with delta potential, demonstrating asymptotic stability of the soliton.

Findings

Proves asymptotic stability of the 1-soliton solution

Extends the method of Bikbaev and Tarasov to the full line

Strengthens earlier results by Holmer and Zworski

Abstract

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski.