A KAM Theorem for Two-dimensional Nonlinear Schr\"odinger Equations

TL;DR

This paper proves an infinite dimensional KAM theorem and applies it to establish the existence of Whitney smooth families of small-amplitude quasi-periodic solutions for a two-dimensional nonlinear Schrödinger equation with specific nonlinearities.

Contribution

The paper introduces a new infinite dimensional KAM theorem and demonstrates its application to a class of 2D nonlinear Schrödinger equations with analytic nonlinearities.

Findings

Existence of Whitney smooth families of quasi-periodic solutions.

Solutions are small-amplitude and partially hyperbolic.

The KAM theorem extends to infinite-dimensional Hamiltonian systems.

Abstract

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two-dimensional nonlinear Schr\"{o}dinger equation $$iu_t-\triangle u +|u|^2u+\frac{\partial{f(x,u,\bar u)}}{\partial{\bar u}}=0, \quad t\in\Bbb R, x\in\Bbb T^2$$ with periodic boundary conditions, where the nonlinearity $\displaystyle f(x,u,\bar u)=\sum_{j,l,j+l\geq6}a_{jl}(x)u^j\bar u^l$, $a_{jl}=a_{lj}$ 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.