# Direct Characterization of Spectral Stability of Small Amplitude Periodic Waves in Scalar Hamiltonian Problems Via Dispersion Relation

## TL;DR

This paper introduces a method to determine the spectral stability of small-amplitude periodic waves in scalar Hamiltonian PDEs by analyzing roots of explicitly constructed polynomials derived from the dispersion relation.

## Contribution

It provides a new explicit polynomial-based criterion for wave stability, linking the dispersion relation to eigenvalue collisions and Krein signature analysis.

## Key findings

- Polynomial roots characterize eigenvalue collisions.
- The method detects stability or instability of small-amplitude waves.
- Application to generalized KdV equations demonstrates effectiveness.

## Abstract

Various approaches to studying the stability of solutions of nonlinear PDEs lead to explicit formulae determining the stability or instability of the wave for a wide range of classes of equations. However, these are typically specialized to a particular equation and checking the stability conditions may not be not straightforward. We present results for a large class of problems that reduce the determination of spectral stability of a wave to a simple task of locating zeros of explicitly constructed polynomials. We study spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems as a perturbation of the zero-amplitude case. A necessary condition for stability of the wave is that the unperturbed spectrum is restricted to the imaginary axis. Instability can come about through a Hamiltonian-Hopf bifurcation, i.e., of a collision of purely imaginary eigenvalues of the Floquet spectrum of opposite Krein signature. In recent work on the stability of small-amplitude waves the dispersion relation of the unperturbed problem was shown to play a central role. We demonstrate that the dispersion relation provides even more explicit information about wave stability: we construct a polynomial of half the degree of the dispersion relation, and its roots directly characterize not only collisions of eigenvalues at zero-amplitude but also an agreement or a disagreement of their Krein signatures. Based on this explicit information it is possible to detect instabilities of non-zero amplitude waves. In our analysis we stay away from the possible instabilities at the origin of the spectral plane corresponding to modulation or Benjamin-Fair instability. Generalized KdV and its higher-order analogues are used as illustrating examples.