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 \Omega_{i}(\xi)=i^d+o(i^{d})+o(i^{\delta}), where $d>0, \delta<0$, 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 Schr\"odinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.