Quasi-Periodic Solutions of Hamiltonian Perturbations of 2D Linear SchroedingerEquations

摘要

This paper is a continuation of the author's work on constructing periodic andquasi-periodic solutions of Hamiltonian perturbations of linear PDE's (with periodic boundary conditions). The method used for this purpose was initiated in the work of Craig and Wayne for 1D-equations and time periodic solutions. It is an infinite dimensional phase space version of the Liapounov-Schmidt argument for the construction of periodic solutions. The basic idea of the Liapounov-Schmidt scheme consists in splitting the problem in a resonant finite dimensional piece given by the Q-equation and the remainder of the problem, the P-equation, which is infinite dimensional and contains the small divisors issues. To achieve frequency variation, we will rely on outer parameters contained in the equation rather than amplitude-frequency modulation depending on the nonlinear term. The model equation considered here is the nonlinear Schrodinger equation.