Characteristic polynomials, related to the normal form of the non linear Schr\"{o}dinger equation

TL;DR

This paper investigates the irreducibility of characteristic polynomials associated with the energy graph of the nonlinear Schrödinger equation, aiming to facilitate the verification of the second Melnikov condition.

Contribution

It introduces a novel analysis of the characteristic polynomial's irreducibility related to the NLS energy graph, aiding in the study of the equation's normal form.

Findings

Proves irreducibility of the characteristic polynomial for NLS energy graph

Provides a method to verify the second Melnikov condition for NLS

Enhances understanding of the algebraic structure of NLS normal form

Abstract

We study the irreducibility of the characteristic polynomial of the energy graph of the non linear Schr\"{o}dinger equation (NLS). This will be useful to the verification of the second Melnikov condition for NLS.