聚焦修正KdV方程的孤子分辨率

IF 1.8 1区数学 Q1 MATHEMATICS, APPLIED

Annales De L Institut Henri Poincare-Analyse Non Lineaire Pub Date : 2021-11-01 DOI:10.1016/j.anihpc.2021.02.008

Gong Chen , Jiaqi Liu

{"title":"聚焦修正KdV方程的孤子分辨率","authors":"Gong Chen , Jiaqi Liu","doi":"10.1016/j.anihpc.2021.02.008","DOIUrl":null,"url":null,"abstract":"<div>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 <math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math>-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory <math><mi>x</mi><mo>=</mo><mtext>v</mtext><mi>t</mi></math> 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.</div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":"38 6","pages":"Pages 2005-2071"},"PeriodicalIF":1.8000,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.02.008","citationCount":"18","resultStr":"{\"title\":\"Soliton resolution for the focusing modified KdV equation\",\"authors\":\"Gong Chen , Jiaqi Liu\",\"doi\":\"10.1016/j.anihpc.2021.02.008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>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 <math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math>-derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory <math><mi>x</mi><mo>=</mo><mtext>v</mtext><mi>t</mi></math> 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.</div>\",\"PeriodicalId\":55514,\"journal\":{\"name\":\"Annales De L Institut Henri Poincare-Analyse Non Lineaire\",\"volume\":\"38 6\",\"pages\":\"Pages 2005-2071\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2021-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.anihpc.2021.02.008\",\"citationCount\":\"18\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales De L Institut Henri Poincare-Analyse Non Lineaire\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0294144921000305\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0294144921000305","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 18

摘要

在一些加权Sobolev空间中，建立了初始条件下聚焦修正Korteweg-de Vries (mKdV)方程的孤子分辨率。我们的方法是基于非线性最陡下降法及其通过∂形式的导数的重新表述。从平稳点的观点出发，我们给出了任意固定v沿轨迹x=vt的精确渐近公式。为了将渐近性推广到低正则性空间中具有初始数据的解，我们通过PDE技术应用了一个全局逼近。作为长渐近性的副产品，我们也得到了包含孤子和呼吸子的非线性结构的渐近稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Soliton resolution for the focusing modified KdV equation

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 $\overline{\partial}$ -derivatives. From the view of stationary points, we give precise asymptotic formulas along trajectory $x = v t$ 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.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Annales De L Institut Henri Poincare-Analyse Non Lineaire 数学-应用数学

CiteScore

4.10

自引率

5.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.