Whitham's equations for modulated roll-waves in shallow flows

Abstract

This paper is concerned with the detailed behaviour of roll-waves undergoing a low-frequency perturbation. We rst derive the so-called Whitham's averaged modulation equations and relate the well-posedness of this set of equations to the spectral stability problem in the small Floquet-number limit. We then fully validate such a system and in particular, we are able to construct solutions to the shallow water equations in the neighbourhood of modulated roll-waves proles that exist for asymptotically large time.

References in corpus (2)

• On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
• Metastability of solitary roll wave solutions of the St. Venant equations with viscosity

Cited by in corpus (7)

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• Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
• Nonlocalized modulation of periodic reaction diffusion waves: Nonlinear stability
• Stability of Periodic Kuramoto-Sivashinsky Waves
• Modulated wave trains in generalized Kuramoto-Sivashinksi equations
• Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit
• Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations