Soliton Resolution for the Derivative Nonlinear Schrodinger Equation

We study the derivative nonlinear Schrodinger 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 (Commun Pure Appl Math 56:1029-1077, 2003) revisited by the -analysis of McLaughlin and Miller (IMRP Int Math Res Pap 48673:1-77, 2006) and Dieng and McLaughlin (Long-time asymptotics for the NLS equation via dbar methods. Preprint, arXiv:0805.2807, 2008), and complemented by the recent work of Borghese etal. (Ann Inst Henri Poincare Anal Non Lineaire, 10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrodinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrodinger equation are asymptotically stable.