{"title":"分数阶Korteweg-de Vries方程的渐近类N孤子解","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":"{\"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}","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}
Asymptotic $N$-soliton-like solutions of the fractional Korteweg–de Vries equation
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
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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.
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