Full-dimensional KAM torus with frequency-preserving in infinite-dimensional Hamiltonian systems

Abstract

In this paper, we present two infinite-dimensional KAM theorems with frequency-preserving for a nonresonant frequency of Diophantine type or even weaker. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the specified frequency independent of any spectral asymptotics, by advantage of the generating function method. This appears to be the first Kolmogorov type result in the infinite-dimensional context. As a direct application, we provide a positive answer to Bourgain's conjecture: full-dimensional invariant tori for 1D nonlinear Schr\"{o}dinger equations do exist.