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

Producción científica: Article › revisión exhaustiva

36 Citas (Scopus)

Resumen

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.

Idioma original	English
Páginas (desde-hasta)	1151-1195
Número de páginas	45
Publicación	Communications in Partial Differential Equations
Volumen	43
N.º	8
DOI	https://doi.org/10.1080/03605302.2018.1475489
Estado	Published - ago 3 2018

Nota bibliográfica

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

Financiación

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.

Financiadores	Número del financiador
National Science Foundation Arctic Social Science Program	DMS-1208778
Simons Foundation	46179-13, 359431

ASJC Scopus subject areas

Analysis
Applied Mathematics

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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.",

keywords = "Derivative nonlinear Schr{\"o}dinger equation, global well-posedness, inverse scattering method, soliton solutions",

author = "Robert Jenkins and Jiaqi Liu and Perry, \{Peter A.\} and Catherine Sulem",

note = "Publisher Copyright: {\textcopyright} 2018, {\textcopyright} 2018 Taylor \& Francis.",

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doi = "10.1080/03605302.2018.1475489",

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AU - Jenkins, Robert

AU - Liu, Jiaqi

AU - Perry, Peter A.

AU - Sulem, Catherine

PY - 2018/8/3

Y1 - 2018/8/3

N2 - 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.

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

UR - https://www.scopus.com/pages/publications/85057341107

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DO - 10.1080/03605302.2018.1475489

M3 - Article

AN - SCOPUS:85057341107

SN - 0360-5302

VL - 43

SP - 1151

EP - 1195

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 8

ER -

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

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ASJC Scopus subject areas

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