{"title":"Stability of a Traveling Wave on a Saddle-Node Trajectory","authors":"L. A. Kalyakin","doi":"10.1134/s0001434624050286","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For semilinear partial differential equations, we consider the solution in the form of a plane wave traveling with a constant velocity. This solution is determined from an ordinary differential equation. A wave that stabilizes at infinity to equilibria corresponds to a phase trajectory connecting fixed points. The fundamental problem of the possibility of using such solutions in applications is their stability in the linear approximation. The stability problem is solved for a wave that corresponds to a trajectory from a saddle to a node. It is known that the velocity is determined ambiguously in this case. In this paper, a method is indicated for finding the limit of the velocity of stable waves for parabolic and hyperbolic equations, which can easily be implemented numerically. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050286","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For semilinear partial differential equations, we consider the solution in the form of a plane wave traveling with a constant velocity. This solution is determined from an ordinary differential equation. A wave that stabilizes at infinity to equilibria corresponds to a phase trajectory connecting fixed points. The fundamental problem of the possibility of using such solutions in applications is their stability in the linear approximation. The stability problem is solved for a wave that corresponds to a trajectory from a saddle to a node. It is known that the velocity is determined ambiguously in this case. In this paper, a method is indicated for finding the limit of the velocity of stable waves for parabolic and hyperbolic equations, which can easily be implemented numerically.
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
