THE DERIVATIVE NONLINEAR SCHR̈ODINGER EQUATION: GLOBAL WELL-POSEDNESS AND SOLITON RESOLUTION

ROBERT JENKINS, JIAQI LIU, PETER PERRY, CATHERINE SULEM

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10 Scopus citations

Abstract

We review recent results on global well-posedness and long-time behavior of smooth solutions to the derivative nonlinear Schr̈odinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [SIAM J. Math. Anal. 20 (1989), pp. 966–986] for treating spectral singularities lead to global well-posedness for general initial conditions in the weighted Sobolev space H2,2pRq. 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.

Original languageEnglish
Pages (from-to)33-73
Number of pages41
JournalQuarterly of Applied Mathematics
Volume78
Issue number1
DOIs
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© (2019) Brown University

ASJC Scopus subject areas

  • Applied Mathematics

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