Periodic Solutions of Hamiltonian Perturbations of 2D Linear SchroedingerEquations

摘要

The purpose of this note is to construct periodic solutions of nonlinear 2DSchroedinger equations, which equations have a parameter corresponding to a perturbation of the linear equation. We consider the case of periodic boundary conditions. Our aim is to study the persistency of time periodic solutions of the linear equation, in the perturbed equation. We follow a method for 1-D equations, which is an infinite dimensional phase space version of the Liapounov-Schmidt argument for the construction of periodic solutions. More recently, the author extended this method to deal also with 1D-quasi periodic problems. In 2D, no such results seem to be known at all, and it seems thus natural to start investigating the periodic case, which is considerably easier.