A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems of infinite dimensions where the second Melnikov’s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schrödinger equations and other equations of any spatial dimensions.