Quasi-periodic solutions for one dimensional Schrödinger equation with quasi-periodic forcing and Dirichlet boundary condition

Abstract

This paper is concerned with a one-dimensional quasi-periodically forced nonlinear Schrödinger equation under Dirichlet boundary conditions. The existence of the quasi-periodic solutions for the equation is verified. By infinitely many symplectic transformations of coordinates, the Hamiltonian of the linear part of the equation can be reduced to an autonomous system. By utilizing the measure estimation of small divisors, there exists a symplectic change of coordinate transformation of the Hamiltonian of the equation into a nice Birkhoff normal form. By an abstract KAM (Kolmogorov-Arnold-Moser) theorem, the existence of a class of small-amplitude quasi-periodic solutions for the above equation is verified.