Quasi-Periodic Solution of Nonlinear Beam Equation on Td with Forced Frequencies

• Zhou Shidi ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
• Idioma: inglés
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• Resumen

  • In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition: utt+Δx2u+Mξu+f(ω¯t,u)=0,u=u(t,x),t∈R,x∈Td,d≥2where Mξ is a real Fourier multiplier, f=f(θ¯,u) is a real analytic function with respect to (θ¯,u), and f(θ¯,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.