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Quasi-Periodic Solution of Nonlinear Beam Equation on $${\mathbb T}^d$$ T d with Forced Frequencies
Qualitative Theory of Dynamical Systems ( IF 1.4 ) Pub Date : 2020-09-17 , DOI: 10.1007/s12346-020-00425-x
Shidi Zhou

In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition:

$$\begin{aligned} u_{tt} + \Delta ^2 _x u + M_{\xi }u + f({\bar{\omega }} t,u)=0,\quad u=u(t,x),t\in {{\mathbb {R}}},x\in {\mathbb T}^d,d\ge 2 \end{aligned}$$

where \(M_{\xi }\) is a real Fourier multiplier, \(f=f({\bar{\theta }},u)\) is a real analytic function with respect to \(({\bar{\theta }},u)\), and \(f({\bar{\theta }},u)=O(|u|^2)\). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori.



中文翻译:

带有强迫频率的$$ {\ mathbb T} ^ d $$ T d上非线性梁方程的拟周期解

在本文中,我们研究周期边界条件下的高维非线性梁方程:

$$ \ begin {aligned} u_ {tt} + \ Delta ^ 2 _x u + M _ {\ xi} u + f({\ bar {\ omega}} t,u)= 0,\ quad u = u(t ,x),t \ in {{\ mathbb {R}}},x \ in {\ mathbb T} ^ d,d \ ge 2 \ end {aligned} $$

其中\(M _ {\ xi} \)是实傅里叶乘法器,\(f = f({\ bar {\ theta}},u)\)是关于\(({ \ theta}},u)\)\(f({\ bar {\ theta}},u)= O(| u | ^ 2)\)。该方程式可以看作是无限维几乎可积分的哈密顿系统。我们建立了无穷维KAM定理,并将其应用到该方程中,以证明存在一类与有限维不变花托相对应的小振幅拟周期解。

更新日期:2020-09-18
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