{"title":"Spectral Stability of Multi-Solitons for the Kaup-Kupershmidt Equation","authors":"Zhong Wang","doi":"10.1007/s00021-025-00942-2","DOIUrl":null,"url":null,"abstract":"<div><p>Spectral stability analysis of ”anomalous” solitons and multi-solitons is presented in the context of a generalized Hamiltonian system called the Kaup-Kupershmidt (KK) equation. The KK equation is a completely integrable fifth order Korteweg-de Vries equation, which admits third order eigenvalue problem in its Lax pair. We also prove Hamiltonian-Krein index identities in verifying stability criterion of its multi-solitons. However, the KK equation does not possess the <span>\\(L^2\\)</span> conservation law and the linearized operators around the multi-solitons have no spectral gap. The main ingredients of the proof are new operator identities for second variation operator and completeness in <span>\\(L^2\\)</span> of the squared eigenfunctions of the third order eigenvalue problem for the KK equation. The operator identities and completeness relation are shown by employing the recursion operators of the KK equation.</p></div>","PeriodicalId":649,"journal":{"name":"Journal of Mathematical Fluid Mechanics","volume":"27 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Fluid Mechanics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00021-025-00942-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Spectral stability analysis of ”anomalous” solitons and multi-solitons is presented in the context of a generalized Hamiltonian system called the Kaup-Kupershmidt (KK) equation. The KK equation is a completely integrable fifth order Korteweg-de Vries equation, which admits third order eigenvalue problem in its Lax pair. We also prove Hamiltonian-Krein index identities in verifying stability criterion of its multi-solitons. However, the KK equation does not possess the \(L^2\) conservation law and the linearized operators around the multi-solitons have no spectral gap. The main ingredients of the proof are new operator identities for second variation operator and completeness in \(L^2\) of the squared eigenfunctions of the third order eigenvalue problem for the KK equation. The operator identities and completeness relation are shown by employing the recursion operators of the KK equation.
期刊介绍：
The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
