{"title":"Asymptotics of Long Nonlinear Propagating Waves in a One-Dimensional Basin with Gentle Shores","authors":"D.S. Minenkov, M.M. Votiakova","doi":"10.1134/S1061920823040143","DOIUrl":null,"url":null,"abstract":"<p> The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom <span>\\(D(x)\\)</span> is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, <span>\\(D(a)=0\\)</span>), or by a shore and a wall. The short-wave asymptotics of the linearized system in the form of a propagating localized wave is constructed. After applying to the constructed functions a simple parametric or explicit change of variables proposed in recent papers (Dobrokhotov, Minenkov, Nazaikinsky, 2022 and Dobrokhotov, Kalinichenko, Minenkov, Nazaikinsky, 2023), we obtain the asymptotics of the original nonlinear problem. On the constructed families of functions, the ratio of the amplitude and the wavelength is studied for which hte wave does not collapse when running up to the shore. </p><p> <b> DOI</b> 10.1134/S1061920823040143 </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"30 4","pages":"621 - 642"},"PeriodicalIF":1.7000,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920823040143","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
The Cauchy problem for a one-dimensional (nonlinear) shallow water equations over a variable bottom \(D(x)\) is considered in an extended basin bounded from two sides by shores (where the bottom degenerates, \(D(a)=0\)), or by a shore and a wall. The short-wave asymptotics of the linearized system in the form of a propagating localized wave is constructed. After applying to the constructed functions a simple parametric or explicit change of variables proposed in recent papers (Dobrokhotov, Minenkov, Nazaikinsky, 2022 and Dobrokhotov, Kalinichenko, Minenkov, Nazaikinsky, 2023), we obtain the asymptotics of the original nonlinear problem. On the constructed families of functions, the ratio of the amplitude and the wavelength is studied for which hte wave does not collapse when running up to the shore.
期刊介绍：
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.
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