The derivative nonlinear Schrödinger equation (DNLS) is the dispersive equation

$ i u_t +u_{xx} = i \epsilon ( |u|^2 u)_x$ where $\epsilon = \pm 1$.

In this talk I will discuss our recent work [4] which uses the complete integrability of DNLS to show that solution of DNLS exist for generic data in the weighted Sobolev space $H^{2,2}(\mathbb{R}) = \{ u \in L^2(\mathbb{R}) \,:\, x^2 u,\ u'' \in L^2(\mathbb{R}) \}$. This class includes data with arbitrary $L^2(\mathbb{R})$ norm and data with finitely many solitons. We use the DBAR generalization of the steepest descent method of Deift and Zhou to show that at large times within space time cones the solution resolves into a finite sum of solitons whose parameters are slowly modulated by their interaction with each other and the background radiation.

We compute an explicit first correction for the dispersive terms. This work builds off the recent work of Liu, Perry, and Sulem [2,3] on DNLS; the DBAR steepest descent analysis is based on the work in [1]. This work is in collaboration with Jiaqi Liu, Peter Perry, and Catherine Sulem.

1. M. Borghese, R. Jenkins, K. McLaughlin. Long time asymptotic behavior of the focusing nonlinear Schrodinger equation. Preprint. arXiv:1604.07436[Math.PH], submitted to Ann. Inst. Henri Poincare C - Analyse non-lineaire.

2. J. Liu, P. Perry, C. Sulem. Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering. Comm. Partial Differential Equations}, {\bf 41} (2016), no. 11, 1692--1760.

3. J. Liu, P. Perry, C. Sulem. Long-time behavior of solutions of the derivative nonlinear Schrödinger equation for soliton-free initial data. Preprint. arXiv:1608.07659[Math.AP], accepted by {\sl Ann. Inst. Henri Poincar\'e C - Analyse non-lin\'eaire}.

4. R. Jenkins, J. Liu, P. Perry, C. Sulem. Global well-posedness and Soliton Resolution for the Derivative Nonlinear Sch\"odinger Equation. Preprint. arXiv:1706.06252[Math.AP].