Linearly stable KAM tori for one dimensional forced Kirchhoff equations with refined Töplitz-Lipschitz property
Linearly stable KAM tori for one dimensional forced Kirchhoff equations with refined Töplitz-Lipschitz property
复制标题
DOI：
10.1016/j.jde.2023.12.041
复制
发表时间：
2024-04
影响因子：
2.4
通讯作者：
Yin Chen;Jiansheng Geng
中科院分区：
文献类型：
--
作者：
Yin Chen;Jiansheng Geng
证明了一个抽象的无限维KAM定理，该定理可用于证明具有Dirichlet边界条件ut t−（1+<$0 π）的一维受迫Kirchhoff方程小振幅拟周期解的存在性和线性稳定性|u x| 2 d x）u x x+ M <$u+<$g（ω <$t，x）= 0，u（t，0）= u（t，π）= 0，其中M <$是真实的傅立叶乘数，g（ω <$t，x）=− g（ω <$t，− x）是真实的解析函数，强迫丢番图频率为ω <$，π是小参数。证明是基于一个改进的Kuksin引理和非对角衰减以及改进的Töplitz-Lipschitz性质的强迫项。
We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with Dirichlet boundary conditions u t t−(1+∫ 0 π| u x| 2 d x) u x x+ M ξ u+ ϵ g (ω¯ t, x)= 0, u (t, 0)= u (t, π)= 0, where M ξ is a real Fourier multiplier, g (ω¯ t, x)=− g (ω¯ t,− x) is real analytic with forced Diophantine frequencies ω¯, ϵ is a small parameter. The proof is based on an improved Kuksin lemma and the off-diagonal decay together with the refined Töplitz-Lipschitz property of the forcing term.