Global Well-posedness and soliton resolution for the Derivative Nonlinear Schrödinger equation

Abstract

We study the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and 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 $\bar{\partial }$-analysis of Dieng-McLaughlin and complemented by the recent work of Borghese-Jenkins-McLaughlin on soliton resolution for the focusing nonlinear Schrödinger equation.

91 pages, 7 figures

References in corpus (2)

• Long-time Asymptotics for the NLS equation via dbar methods
• Long-Time Behavior of Solutions to the Derivative Nonlinear Schrödinger Equation for Soliton-Free Initial Data

Cited by in corpus (1)

• Global Well-Posesedness for the Derivative Nonlinear Schrodinger Equation