Quasi-periodic solutions for a Schr\xf6dinger equation with a quintic nonlinear term depending on the time and space variables

Yi Wang,Jie Liu,Min Zhang

doi:10.1186/s13661-018-0996-9

Abstract

This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.

Highlights

IntroductionA Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term iut – uxx + mu + εg(ωt, x)|u|4u = 0, (1)
In this paper, a Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term iut – uxx + mu + εg(ωt, x)|u|4u = 0, (1)under the Dirichlet boundary conditions u(t, 0) = u(t, π) = 0 (2)is considered, where mis real; ε is a small positive parameter; ω ∈ [, 2 ]κ ( > 0) is a κ-dimensional frequency vector; κ ≥ 1 is an integer; and the function g(ωt, x) = g(θ, x), (θ, x) ∈ Tκ × [0, π] is real analytic in (θ, x) and quasi-periodic in t
We aim to explore whether the boundary value problem (1) with (2) has real analytic, linearly stable, and quasi-periodic solutions

Summary

Introduction

A Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term iut – uxx + mu + εg(ωt, x)|u|4u = 0, (1). Is considered, where mis real; ε is a small positive parameter; ω ∈ [ , 2 ]κ ( > 0) is a κ-dimensional frequency vector; κ ≥ 1 is an integer; and the function g(ωt, x) = g(θ, x), (θ, x) ∈ Tκ × [0, π] is real analytic in (θ, x) and quasi-periodic in t. We aim to explore whether the boundary value problem (1) with (2) has real analytic, linearly stable, and quasi-periodic solutions. We study this equation as an infinite-dimensional Hamiltonian system. One is the Craig– Wayne–Bourgain (CWB) method and the other is the infinite-dimensional KAM theory. The KAM method can capture more properties of quasi-periodic solutions such as their

Objectives

Conclusion