Soliton resolution for the focusing modified KdV equation

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-11-01 DOI:10.1016/j.anihpc.2021.02.008

Gong Chen , Jiaqi Liu

引用次数: 18

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 $\overline{\partial}$ -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.

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聚焦修正KdV方程的孤子分辨率

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。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.