KAM tori for the two-dimensional completely resonant Schrödinger equation with the general nonlinearity

Abstract

In this paper, two-dimensional completely resonant Schrödinger equation with the general nonlinearityiut−Δu+|u|2pu=0,x∈T2:=R2/(2πZ)2,t∈R,p∈Z+ under periodic boundary conditions is considered. For any given positive integers p and b, it is obtained that a Whitney smooth family of small-amplitude b-quasi-periodic solutions for the equation by developing an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The essence of the normal form is the integrable part of the Hamiltonian system after introducing the action-angle variable, which contains both the linear integrable part of the Schrödinger equation and the integrable part from the nonlinearity. Determining such normal form, in general, is not easy task as it requires some novel ideas and large number of complex calculations. This work includes some new ideas and overcomes some technical difficulties, which solves more completely the existence problem of quasi-periodic solutions of the completely resonant Schrödinger equation on the two dimensional torus with the general nonlinearity.