{"title":"Nonisospectral equations from the Cauchy matrix approach","authors":"Alemu Yilma Tefera, Shangshuai Li, Da-jun Zhang","doi":"10.1016/S0034-4877(24)00055-7","DOIUrl":null,"url":null,"abstract":"<div><div>The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {<em>S</em><sup>(<em>i,j</em>)</sup>} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {<em>S</em><sup>(<em>i,j</em>)</sup>} can be derived. Some identities of {<em>S</em><sup>(<em>i,j</em>)</sup>} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 47-72"},"PeriodicalIF":1.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487724000557","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
The Cauchy matrix approach is developed to construct explicit solutions for some nonisospectral equations, including the nonisospectral Korteweg–de Vries (KdV) equation, the nonisospectral modified KdV equation, and the nonisospectral sine-Gordon equation. By means of a Sylvester equation, a set of scalar master functions {S(i,j)} is defined. We show how nonisospectral dispersion relations are introduced such that the evolutions of {S(i,j)} can be derived. Some identities of {S(i,j)} are employed in verifying solutions. Some explicit one-soliton and two-soliton solutions are illustrated together with analysis of their dynamics.
期刊介绍:
Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.