# A method for identifying stability regimes using roots of a reduced-order polynomial

## TL;DR

This paper introduces a polynomial-based method to determine the stability of small-amplitude periodic travelling waves in dispersive Hamiltonian PDEs by analyzing the roots of a reduced-order polynomial derived from the dispersion relation and Hamiltonian signature.

## Contribution

It develops a novel approach combining two stability criteria into a single polynomial, enabling straightforward stability analysis through root examination.

## Key findings

- The polynomial contains no real roots for stable solutions.
- The method simplifies stability analysis by reducing it to root-finding.
- Example analysis demonstrates the effectiveness of the approach.

## Abstract

For dispersive Hamiltonian partial differential equations of order 2N+1, N integer, there are two criteria to analyse to examine the stability of small-amplitude, periodic travelling wave solutions to high-frequency perturbations. The first necessary condition for instability is given via the dispersion relation. The second criterion for instability is the signature of the eigenvalues of the spectral stability problem given by the sign of the Hamiltonian. In this work, we show how to combine these two conditions for instability into a polynomial of degree N. If the polynomial contains no real roots, then the travelling wave solutions are stable. We present the method for deriving the polynomial and analyse its roots using Sturm's theory via an example.