{"title":"Quasi-periodic swing via weak KAM theory","authors":"Xun Niu , Kaizhi Wang , Yong Li","doi":"10.1016/j.physd.2025.134559","DOIUrl":null,"url":null,"abstract":"<div><div>Our primary focus is on the study of the dynamics of quasi-periodic swing equations from the weak KAM point of view. To achieve this, we initially explore a class of quasi-periodic Hamiltonian systems. We discover that a limit function, derived from the convergence of a sequence of functional minimizers, satisfies the Hamilton–Jacobi equations in the context of minimal measures. This is the so-called weak KAM solution. Subsequently, we establish the existence of invariant torus for the swing equation in a weak sense. Lastly, we discuss certain properties of the weak KAM solution for a one-dimensional periodic swing equation.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"474 ","pages":"Article 134559"},"PeriodicalIF":2.7000,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925000387","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Our primary focus is on the study of the dynamics of quasi-periodic swing equations from the weak KAM point of view. To achieve this, we initially explore a class of quasi-periodic Hamiltonian systems. We discover that a limit function, derived from the convergence of a sequence of functional minimizers, satisfies the Hamilton–Jacobi equations in the context of minimal measures. This is the so-called weak KAM solution. Subsequently, we establish the existence of invariant torus for the swing equation in a weak sense. Lastly, we discuss certain properties of the weak KAM solution for a one-dimensional periodic swing equation.
期刊介绍：
Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.
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