A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

Abstract

In this paper, one-dimensional ( 1 D ) nonlinear Schrödinger equation iu t - u xx + mu + ∂ g ( u , u ¯ ) ∂ u ¯ = 0 , with Periodic Boundary Conditions is considered; m ∉ 1 12 Z is a real parameter and the nonlinearity g ( u , u ¯ ) = ∑ j , l , j + l ⩾ 4 a jl u j u ¯ l , a jl = a lj ∈ R , a 22 ≠ 0 is a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system.