KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics Ω_{i}(ξ)=i^d+o(i^{d})+o(i^δ), where $d>0, δ<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schrödinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.