Soliton Resolution for the Derivative Nonlinear Schr\"odinger Equation

TL;DR

This paper analyzes the long-time behavior of solutions to the Derivative Nonlinear Schrödinger equation, showing they decompose into multi-solitons and dispersive waves, with explicit correction terms, using advanced steepest descent and $ar{ ext{d}}$-methods.

Contribution

It provides a comprehensive soliton resolution result for the Derivative Nonlinear Schrödinger equation, including explicit asymptotics and stability analysis, extending previous work to this specific equation.

Findings

Solutions decompose into multi-solitons and dispersive parts

Explicit formulas for correction dispersive terms

Asymptotic stability of N-soliton solutions

Abstract

We study the Derivative Nonlinear Schr\"odinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton-soliton and soliton-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou, revisited by the $\overline{\partial}$-analysis of {McLaughlin-Miller and Dieng-McLaughlin, and complemented by the recent work of Borghese-Jenkins-McLaughlin on soliton resolution for the focusing Nonlinear Schr\"odinger equation. Our results imply that $N$-soliton solutions of the Derivative Nonlinear Schr\"odinger equation are asymptotically stable.