Global well-posedness for the derivative non-linear Schrödinger equation

Robert Jenkins; Jiaqi Liu; Peter A. Perry; Catherine Sulem

doi:10.1080/03605302.2018.1475489

Global well-posedness for the derivative non-linear Schrödinger equation

Robert Jenkins, Jiaqi Liu, Peter A. Perry, Catherine Sulem

Mathematics

Research output: Contribution to journal › Article › peer-review

22 Scopus citations

Abstract

We study the derivative nonlinear Schrödinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large L ² -norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of N-soliton solutions.

Original language	English
Pages (from-to)	1151-1195
Number of pages	45
Journal	Communications in Partial Differential Equations
Volume	43
Issue number	8
DOIs	https://doi.org/10.1080/03605302.2018.1475489
State	Published - Aug 3 2018

Bibliographical note

Publisher Copyright:
© 2018, © 2018 Taylor & Francis.

Funding

PAP was supported in part by NSF Grant DMS-1208778 and by Simons Foundation Research and Travel Grant 359431, and CS was supported in part by Grant 46179-13 from the Natural Sciences and Engineering Research Council of Canada.

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS-1208778
National Science Foundation Arctic Social Science Program
Simons Foundation	46179-13, 359431
Simons Foundation
Natural Sciences and Engineering Research Council of Canada

Keywords

Derivative nonlinear Schrödinger equation
global well-posedness
inverse scattering method
soliton solutions

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/03605302.2018.1475489

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title = "Global well-posedness for the derivative non-linear Schr{\"o}dinger equation",

abstract = " We study the derivative nonlinear Schr{\"o}dinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large L 2 -norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of N-soliton solutions.",

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AU - Liu, Jiaqi

AU - Perry, Peter A.

AU - Sulem, Catherine

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Y1 - 2018/8/3

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AB - We study the derivative nonlinear Schrödinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large L 2 -norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of N-soliton solutions.

KW - Derivative nonlinear Schrödinger equation

KW - global well-posedness

KW - inverse scattering method

KW - soliton solutions

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JF - Communications in Partial Differential Equations

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Global well-posedness for the derivative non-linear Schrödinger equation

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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