Travelling wave solutions for some two-component shallow water models

Denys Dutykh (1, 2, 3) , Delia Ionescu-Kruse (4)

1 LAMA - Laboratoire de Mathématiques
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 IMAR - "Simion Stoilow" Institute of Mathematics

In the present study we perform a unified analysis of travelling wave solutions to three different two-component systems which appear in shallow water theory. Namely, we analyse the celebrated Green-Naghdi equations, the integrable two-component Camassa-Holm equations and a new two-component system of Green-Naghdi type. In particular, we are interested in solitary and cnoidal-type solutions, as two most important classes of travelling waves that we encounter in applications. We provide a complete phase-plane analysis of all possible travelling wave solutions which may arise in these models. In particular, we show the existence of new type of solutions.

Mots clés: travelling waves; cnoidal waves; solitary waves; phase-plane analysis; Camassa-Holm equations; Green-Naghdi model; Serre equations

Domaines: Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]; Physique Atmosphérique et Océanique [physics.ao-ph]; Dynamique des Fluides [physics.flu-dyn]; Systèmes Solubles et Intégrables [nlin.SI]; Formation de Structures et Solitons [nlin.PS]

Denys Dutykh, Delia Ionescu-Kruse. Travelling wave solutions for some two-component shallow water models. Journal of Differential Equations, 2016, 261 (2), pp.1099-1114. ⟨10.1016/j.jde.2016.03.035⟩. ⟨hal-01294603⟩