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

Pascal Noble (1) , Luis Miguel Miguel Rodrigues (1)

1 ICJ - Institut Camille Jordan

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.

Mots clés: modulation; wave trains; periodic travelling waves; Saint-Venant equations; Bloch decomposition

Domaines: Mathématiques générales [math.GM]; Equations aux dérivées partielles [math.AP]

Pascal Noble, Luis Miguel Miguel Rodrigues. Whitham's equations for modulated roll-waves in shallow flows. 2010. ⟨hal-00534173⟩