2017/10/31 by Robert Jenkins, Jiaqi Liu, Peter Perry +1 · 3 citations
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.MP #math.SP #msc:35Q55 #msc:37K15 #msc:37K40 #msc:35A01 #msc:35P25
paper · pdf · doi:10.1007/s00220-018-3138-4
44 pages, 4 figures. This article is a revision of sections 5-7 and appendices A and C of arXiv:1706.06252. The larger paper has been split into two articles. This version is the final version to appear in Comm. Math. Phys. and incorporates a number of suggestions by the referee
arxiv created 2018/02/17 · arxiv updated 2018/05/23
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, revisited by the ∂-analysis of McLaughlin-Miller and Dieng-McLaughlin, and complemented by the recent work of Borghese-Jenkins-McLaughlin 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.