Floquet theory and stability for Hamiltonian partial differential equations

TL;DR

This paper applies Floquet theory to analyze the stability of periodic traveling waves in various Hamiltonian PDEs, revealing spectral symmetries and bifurcations through analytical and numerical methods.

Contribution

It introduces a symmetry-based approach to determine the spectrum and bifurcations of Hamiltonian PDEs using Floquet theory, supported by numerical evidence.

Findings

Characteristic polynomial inherits symmetry from PDEs

Essential spectrum along the imaginary axis identified

Bifurcations of spectrum away from the axis observed

Abstract

We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schr\"odinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.