The Derivative Nonlinear Schr\"{o}dinger Equation: Global Well-Posedness and Soliton Resolution

Abstract

We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schr\"{o}dinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou for treating spectral singularities lead to global wellposedness for general initial conditions in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.