Reducible quasi-periodic solutions for the Non Linear Schr\"odinger equation

TL;DR

This paper constructs small reducible quasi-periodic solutions for the resonant nonlinear Schrödinger equation on multi-dimensional tori, demonstrating reducibility, block-diagonality, and Melnikov conditions via a KAM approach.

Contribution

It proves the reducibility of the normal form for the resonant NLS, enabling the construction of quasi-periodic solutions on higher-dimensional tori.

Findings

Normal form is reducible and block diagonal.

Normal form satisfies the second Melnikov condition.

KAM algorithm yields quasi-periodic solutions.

Abstract

The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a $d$--dimensional torus $\T^d$. The main point is to prove that prove that the normal form is reducible, block diagonal and satisfies the second Melnikov condition block wise. From this we deduce the result by a KAM algorithm.