Asymptotic $N$-soliton-like solutions of the fractional Korteweg–de Vries equation

IF 1.3 2区数学 Q1 MATHEMATICS Revista Matematica Iberoamericana Pub Date : 2021-12-21 DOI:10.4171/rmi/1396

Arnaud Eychenne

{"title":"Asymptotic $N$-soliton-like solutions of the fractional Korteweg–de Vries equation","authors":"Arnaud Eychenne","doi":"10.4171/rmi/1396","DOIUrl":null,"url":null,"abstract":"We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \\partial_t u - \\partial_x\\left(|D|^{\\alpha}u - u^2 \\right)=0, $$ in the whole sub-critical range $\\alpha \\in]\\frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0<c_1<\\cdots<c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $$ \\lim_{t\\to\\infty} \\| U(t,\\cdot) - \\sum_{j=1}^NQ_{c_j}(x-\\rho_j(t)) \\|_{H^{\\frac{\\alpha}2}}=0, $$ where $\\rho'_j(t) \\sim c_j$ as $t \\to +\\infty$. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state $Q_c$ and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Lin\\'eaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Revista Matematica Iberoamericana","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/rmi/1396","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 3

Abstract

We construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $$ \partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0, $$ in the whole sub-critical range $\alpha \in]\frac12,2[$. More precisely, if $Q_c$ denotes the ground state solution associated to fKdV evolving with velocity $c$, then given $0

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

分数阶Korteweg-de Vries方程的渐近类N孤子解

我们构造了分数阶Korteweg-de Vries (fKdV)方程$$ \partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0, $$在整个亚临界范围$\alpha \in]\frac12,2[$的$N$ -孤子解。更准确地说，如果$Q_c$表示与fKdV随速度$c$演化相关的基态解，则给定$0

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Revista Matematica Iberoamericana 数学-数学

CiteScore

2.40

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Revista Matemática Iberoamericana publishes original research articles on all areas of mathematics. Its distinguished Editorial Board selects papers according to the highest standards. Founded in 1985, Revista is a scientific journal of Real Sociedad Matemática Española.

期刊最新文献

The Poincaré problem for reducible curves Mordell–Weil groups and automorphism groups of elliptic $K3$ surfaces A four-dimensional cousin of the Segre cubic Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane Jet spaces over Carnot groups