A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. (English) Zbl 1103.37047
Summary: We give a KAM theorem for a class of infinite-dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher-dimensional spaces with periodic boundary conditions to construct linearly stable quasi-periodic solutions and its local Birkhoff normal form. The applications to the higher-dimensional beam equations and the higher-dimensional Schrödinger equations with nonlocal smooth nonlinearity are given in this paper, too.
MSC:
37K55 | Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems |
35Q72 | Other PDE from mechanics (MSC2000) |
35Q55 | NLS equations (nonlinear Schrödinger equations) |
35L75 | Higher-order nonlinear hyperbolic equations |
Keywords:
infinite-dimensional nearly integrable Hamiltonian systems; quasi-periodic solutions; Birkhoff normal form; higher-dimensional beam equations; higher-dimensional Schrödinger equationsReferences:
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