On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

2019/03/01 by Hongzi Cong, Cong, Hongzi, Lufang Mi +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Photonic Systems #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1903.00127

openalex publication_date 2019/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the following nonlinear Schrödinger equation iu_t-u_xx+V*u+εf(x)|u|⁴u=0, x∈\mathbbT=ℝ/2πℤ, where V* is the Fourier multiplier defined by \widehat(V* u)_n=V_n\widehatu_n, V_n∈[-1,1] and f(x) is Gevrey smooth. It is shown that for 0≤|ε|≪1, there is some (V_n)_n∈ℤ such that, the equation admits a time almost periodic solution (i.e., full dimensional KAM torus) in the Gevrey space. This extends results of Bourgain \citeBJFA2005 and Cong-Liu-Shi-Yuan \citeCLSY to the case that the nonlinear perturbation depends explicitly on the space variable x. The main difficulty here is the absence of zero momentum of the equation.

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