QUASI-PERIODIC SOLUTIONS OF 1D NONLINEAR SCHRÖDINGER EQUATION WITH A MULTIPLICATIVE POTENTIAL

Abstract

This paper deals with one-dimensional (1D) nonlinear Schrödinger equation with a multiplicative potential, subject to Dirichlet boundary conditions. It is proved that for each prescribed integer $b\gt 1$, the equation admits small-amplitude quasi-periodic solutions, whose $b$-dimensional frequencies are small dilation of a given Diophantine vector. The proof is based on a modified infinite-dimensional KAM theory.