On the existence of full dimensional KAM tori for 1D periodic nonlinear Schrödinger equation
Yuan Wu
TL;DR
The paper proves the existence of a full-dimensional KAM torus for the 1D periodic nonlinear Schrödinger equation with a Fourier multiplier and a space-dependent Gevrey nonlinearity, under a sub-exponential decay envelope for the action variables $I_n$. Employing a rapidly convergent Newton-type KAM scheme in Cartesian coordinates, the authors solve homological equations under a Diophantine condition on the frequency vector and carefully control small divisors through bespoke Gevrey-analytic norms. They construct a sequence of near-identity symplectic transformations that yield a final invariant torus with prescribed frequencies $\big(n^2+\omega_n\big)$ and linear stability, with amplitudes localized by $I_n$ in a Gevrey-decay band. This work extends Bourgain’s full-dimensional tori results to cases where the nonlinear perturbation depends explicitly on the spatial variable, using a slower decay rate $I_n \sim e^{-2\ln^{\sigma}|n|}$ ($\sigma>2$).
Abstract
In this paper, we will prove the existence of full dimensional tori for 1-dimensional nonlinear Schrödinger equation \begin{eqnarray}\label{maineq0} \mathbf{i}u_{t}-u_{xx}+V*u+εf(x)|u|^{4}u=0,\ x\in\mathbb{T}=\mathbb{R}/2π\mathbb{Z}, \end{eqnarray} with boundary conditions, where $V*$ is the Fourier multiplier, and $f(x)$ is Gevrey smooth. Here the radius of the invariant tori satisfies a slower decay, i.e. \[ I_n\sim e^{-2\ln^σ|n|}, \mbox{as}\ n\rightarrow\infty, \] for any $ σ> 2, $ which extends results of Bourgain \cite{BJFA2005} and Cong \cite{cong2024} to the case that the nonlinear perturbation depends explicitly on the space variable $x$.