Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions
Yin Chen, Jiansheng Geng, Guangzhao Zhou
公開日: 2025/9/5
Abstract
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 periodic boundary conditions \[ u_{tt}-(1+\int_{0}^{2\pi} |u_x|^2 dx)u_{xx}+ M_\xi u+\epsilon g(\bar{\omega}t,x) =0,\quad u(t,x+2\pi)=u(t,x),\] where $M_\xi$ is a real Fourier multiplier, $g(\bar{\omega}t,x)$ is real analytic with forced Diophantine frequencies $\bar\omega$, $\epsilon$ is a small parameter. The paper generalizes the previous results from the simple eigenvalue to the double eigenvalues under the quasi-linear perturbation.