Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

Jiaqi Liu; Peter A. Perry; Catherine Sulem

doi:10.1080/03605302.2016.1227337

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

Jiaqi Liu, Peter A. Perry, Catherine Sulem

Mathematics

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

41 Scopus citations

Abstract

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.

Original language	English
Pages (from-to)	1692-1760
Number of pages	69
Journal	Communications in Partial Differential Equations
Volume	41
Issue number	11
DOIs	https://doi.org/10.1080/03605302.2016.1227337
State	Published - Nov 1 2016

Bibliographical note

Publisher Copyright:
© 2016, Copyright © Taylor & Francis Group, LLC.

Keywords

Derivative nonlinear Schrodinger equation
global well-posedness
inverse scattering method
solitonless solutions

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/03605302.2016.1227337

Cite this

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title = "Global existence for the derivative nonlinear Schr{\"o}dinger equation by the method of inverse scattering",

abstract = "We develop inverse scattering for the derivative nonlinear Schr{\"o}dinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schr{\"o}dinger equation.",

keywords = "Derivative nonlinear Schrodinger equation, global well-posedness, inverse scattering method, solitonless solutions",

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AU - Liu, Jiaqi

AU - Perry, Peter A.

AU - Sulem, Catherine

PY - 2016/11/1

Y1 - 2016/11/1

N2 - We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.

AB - We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.

KW - Derivative nonlinear Schrodinger equation

KW - global well-posedness

KW - inverse scattering method

KW - solitonless solutions

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JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

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Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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