Metastability of solitary roll wave solutions of the St. Venant equations with viscosity

TL;DR

This paper investigates the stability of solitary wave solutions in viscous shallow-water equations, revealing metastability with stable point spectrum but unstable essential spectrum, and proposes a wave interaction mechanism.

Contribution

It introduces the concept of metastable solitary waves with mixed spectral stability and explores their stabilization through wave interactions, supported by numerical and analytical methods.

Findings

Identification of metastable solitary waves with stable point spectrum

Demonstration of convective instabilities shed in the wave wake

Numerical evidence of possible stable periodic solutions near homoclinic orbits

Abstract

We study by a combination of numerical and analytical Evans function techniques the stability of solitary wave solutions of the St. Venant equations for viscous shallow-water flow down an incline, and related models. Our main result is to exhibit examples of metastable solitary waves for the St. Venant equations, with stable point spectrum indicating coherence of the wave profile but unstable essential spectrum indicating oscillatory convective instabilities shed in its wake. We propose a mechanism based on ``dynamic spectrum'' of the wave profile, by which a wave train of solitary pulses can stabilize each other by de-amplification of convective instabilities as they pass through successive waves. We present numerical time evolution studies supporting these conclusions, which bear also on the possibility of stable periodic solutions close to the homoclinic. For the closely related viscous Jin-Xin model, by contrast, for which the essential spectrum is stable, we show using the stability index of Gardner--Zumbrun that solitary wave pulses are always exponentially unstable, possessing point spectra with positive real part.