A KAM Theorem for Two-dimensional Nonlinear Schrödinger Equations

2019/09/06 by Geng, Jiansheng, Xue, Shuaishuai
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.02681

Abstract

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schrödinger equation iu_t-\triangle u +|u|²u+\frac∂f(x,u, u)∂ u=0, t∈\Bbb R, x∈\Bbb T² with periodic boundary conditions, where the nonlinearity f(x,u, u)=∑_j,l,j+l≥6a_jl(x)u^j u^l, a_jl=a_lj is a real analytic function in a neighborhood of the origin. We obtain for the equation a Whitney smooth family of small--amplitude quasi--periodic solutions which are partially hyperbolic.

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