Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

We study the derivative nonlinear Schrödinger 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 et al. (Ann Inst Henri Poincaré Anal Non Linéaire, https://doi.org/10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrödinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrödinger equation are asymptotically stable.