Soliton resolution for the focusing modified KdV equation

Gong Chen; Jiaqi Liu

doi:10.1016/j.anihpc.2021.02.008

Soliton resolution for the focusing modified KdV equation

Gong Chen
, Jiaqi Liu

Mathematics

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

38 Scopus citations

Abstract

The soliton resolution for the focusing modified Korteweg-de Vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through ∂‾-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory x=vt for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.

Original language	English
Pages (from-to)	2005-2071
Number of pages	67
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	38
Issue number	6
DOIs	https://doi.org/10.1016/j.anihpc.2021.02.008
State	Published - Nov 1 2021

Bibliographical note

Publisher Copyright:
© 2021 L'Association Publications de l'Institut Henri Poincaré

Funding

We want to thank Prof. Catherine Sulem for her many useful comments.

Keywords

Breather stability
Long time asymptotics
Riemann-Hilbert problems
Soliton resolution

ASJC Scopus subject areas

Analysis
Mathematical Physics
Applied Mathematics

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10.1016/j.anihpc.2021.02.008

Cite this

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abstract = "The soliton resolution for the focusing modified Korteweg-de Vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through ∂‾-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory x=vt for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.",

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AU - Liu, Jiaqi

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AB - The soliton resolution for the focusing modified Korteweg-de Vries (mKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method and its reformulation through ∂‾-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory x=vt for any fixed v. To extend the asymptotics to solutions with initial data in low regularity spaces, we apply a global approximation via PDE techniques. As by-products of our long-time asymptotics, we also obtain the asymptotic stability of nonlinear structures involving solitons and breathers.

KW - Breather stability

KW - Long time asymptotics

KW - Riemann-Hilbert problems

KW - Soliton resolution

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DO - 10.1016/j.anihpc.2021.02.008

M3 - Article

AN - SCOPUS:85101568794

SN - 0294-1449

VL - 38

SP - 2005

EP - 2071

JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 6

ER -

Soliton resolution for the focusing modified KdV equation

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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