Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

TL;DR

This paper investigates the spectral and nonlinear stability of periodic traveling waves in hyperbolic-parabolic systems, using Whitham averaged equations, with analytical and numerical evidence for stability in certain regimes.

Contribution

It introduces spectral stability assumptions based on Whitham equations and provides a proof of nonlinear stability, along with numerical verification of stability regimes.

Findings

Spectral instability in Hopf and homoclinic limits.

Numerical evidence of spectral stability at intermediate periods.

Proposed mechanism for metastability of limiting orbits.

Abstract

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.