{"title":"Orbital stability of smooth solitons in H1 ∩ W1,4 for the modified Camassa-Holm equation","authors":"Qian Zhang, Guangming Zhu","doi":"10.1016/j.jde.2024.10.032","DOIUrl":null,"url":null,"abstract":"<div><div>We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∩</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msup></math></span> stability result in Li and Liu (2021) <span><span>[8]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1390-1404"},"PeriodicalIF":2.4000,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624006879","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the stability result in Li and Liu (2021) [8].
我们分析了修正的卡马萨-霍尔姆方程中光滑孤波的稳定性，该方程是一种准线性、可积分的浅水波传播模型。通过相位肖像分析，我们确定了在一定分散参数范围内的唯一平滑孤波。利用变分法，我们证明了这种波在小扰动下的轨道稳定性，求解了一个带约束条件的最小化问题。我们加强了 Li 和 Liu (2021) [8] 中的 H1∩W1,4 稳定性结果。
期刊介绍：
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics
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