The energy graph of the non linear Schr\"odinger equation

TL;DR

This paper proves algebraic, combinatorial, and geometric theorems crucial for establishing the stability of certain solutions to the nonlinear Schrödinger equation on a torus, advancing understanding of its normal forms.

Contribution

It provides new algebraic and combinatorial proofs essential for analyzing the stability of specific NLS solutions on a torus.

Findings

Proved theorems of algebraic, combinatorial, and geometric nature.

Established stability criteria for certain NLS solutions.

Enhanced the theoretical framework for analyzing nonlinear Schrödinger equations.

Abstract

We discuss the stability of a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus described in a previous paper. The discussion is essentially combinatorial and algebraic in nature. Thus this paper contains the proof of two Theorems of algebraic, combinatorial and geometric nature, which we need in order to prove stability of certain solutions of the non--linear Schr\"odinger (NLS) equation on a torus.