Quasi-Periodic Solution of Nonlinear Beam Equation on \({\mathbb T}^d\) with Forced Frequencies

Abstract

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.

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Correspondence to Shidi Zhou.

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Zhou, S. Quasi-Periodic Solution of Nonlinear Beam Equation on \({\mathbb T}^d\) with Forced Frequencies. Qual. Theory Dyn. Syst. 19, 89 (2020). https://doi.org/10.1007/s12346-020-00425-x

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Keywords

  • KAM theory
  • Hamiltonian systems
  • Beam equation
  • Birkhoff normal form

Mathematics Subject Classification

  • Primary 37K55, 35B10