Spectral validation of the whitham equations for periodic waves of lattice dynamical systems

TL;DR

This paper rigorously validates the Whitham modulation equations for periodic waves in lattice dynamical systems, demonstrating their spectral stability conditions and linking modulation velocities to group velocities using discrete Bloch transform techniques.

Contribution

It provides the first simultaneous proof for reaction-diffusion, balance laws, and Hamiltonian lattice systems, extending continuous spectral stability results to semi-discrete systems.

Findings

Weak hyperbolicity is necessary for spectral stability of waves.

Characteristic velocities of the modulation system match group velocities when weak hyperbolicity holds.

Discrete Bloch transform effectively analyzes spectral properties of lattice systems.

Abstract

In the present contribution we investigate some features of dynamical lattice systems near periodic traveling waves. First, following the formal averaging method of Whitham, we derive modulation systems expected to drive at main order the time evolution of slowly modulated wavetrains. Then, for waves whose period is commensurable to the lattice, we prove that the formally-derived first-order averaged system must be at least weakly hyperbolic if the background waves are to be spectrally stable, and, when weak hyperbolicity is met, the characteristic velocities of the modulation system provide group velocities of the original system. Historically, for dynamical evolutions obeying partial differential equations, this has been proved, according to increasing level of algebraic complexity, first for systems of reaction-diffusion type, then for generic systems of balance laws, at last for Hamiltonian systems. Here, for their semi-discrete counterparts, we give at once simultaneous proofs for all these cases. Our main analytical tool is the discrete Bloch transform, a discrete analogous to the continuous Bloch transform. Nevertheless , we needed to overcome the absence of genuine space-translation invariance, a key ingredient of continuous analyses.