The quasi-periodic solution of fractional nonlinear Schrödinger equation on tori

Abstract

This paper is concerned with the fractional nonlinear Schrödinger equation with periodic boundary conditions iut+−Δs0u+−Δr0f(|u|2)u=0,x∈T≔R/2πZ,where s0∈(1/2,1), r0∈[0,s0−1/2] and f is a real analytic function in some neighborhood of the origin in ℂ. The above equation (0.1) is reversible with respect to the involution G:(u,ū)↦(ū,u). By the KAM (Kolmogorov–Arnold–Moser) theorem for an infinite dimensional reversible system with unbounded perturbation, we obtain that there exists Cantor families of quasi-periodic solutions of small amplitude to above equation.