Catherine Sulem (University of Toronto)

Global Wellposedness and Soliton Resolution for Derivative Nonlinear Schrödinger equation

We prove global wellposedness of the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces. For initial conditions that support bright solitons (but exclude spectral singularities), 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. Our analysis provides explicit formulae for the multi-soliton component as well as the correction dispersive term. Our approach exploits the complete integrability of DNLS. We use techniques of inverse scattering and the nonlinear steepest descent method (Zhou 1989, Deift-Zhou 1993) revisited by the ∂-analysis of Dieng-McLaughlin (2008) and complemented by the recent work of Borghese-Jenkins-McLaughlin (2017) on soliton resolution for the focusing nonlinear Schrödinger equation. This is a joint work with Robert Jenkins, Jiaqi Liu and Peter Perry.