摘要
基于非线性薛定谔方程在系统遇到强迫力时,动力学行为会产生影响,通过带有无穷维频率小除数的正规形方法,利用概周期强迫的KAM定理,得到了方程的不变环面的存在性.首先给出无穷维频率的非共振条件,引入作用量角变量,得到非线性薛定谔方程的哈密顿函数,通过构造典则变换,接着利用文[12]中的无穷维频率的小除数概率测度估计方法,然后通过提取系统的正规形来验证系统满足无穷维KAM定理的条件,最后得到系统的概周期解.结果表明,在系统受到满足一定条件的概周期强迫力时,大部分不变环面能得以保持,且仅有微小形变.
This paper focuses on a nonlinear Schrödinger equation with almost-periodic forc-ing under periodic boundary conditions.By utilizing the Diophantine conditions of infinitely dimensional small divisors,and introducing a pair of action-angle variables,we obtain the Hamiltonian equations.Applying the measure estimates of the infinitely dimensional small divisors,we construct a symplectic change of coordinates which transforms the Hamiltonian into the partial Birkhoff normal form.It is proved that the our Hamiltonian satisfies the conditions of the KAM theorem for the Hamiltonian systems with almost-periodic forcing.Thus,the nonlinear Schrödinger equation admits the almost-periodic solutions with most values of the infinite frequency vector.
作者
芮杰
江一凡
李垚彤
陈显斌
陈骏杰
RUI Jie;JIANG Yi-fan;LI Yao-tong;CHEN Xian-bin;CHEN Jun-jie(College of Science,China Universityof Petroleum,Qingdao 266580,China)
出处
《数学的实践与认识》
北大核心
2025年第11期242-247,共6页
Mathematics in Practice and Theory
基金
山东省自然科学基金面上项目(ZR2022MA078)
国家自然科学基金青年项目(61902429)
2024年山东省研究生优质教育教学资源项目(SDYAL2024041)。
关键词
概周期强迫
不变环面
薛定谔方程
KAM理论
almost-periodic forcing
invariant tori
Schrödinger equations
KAM theory
