Effects of vorticity on the travelling waves of some shallow water two-component systems

Denys Dutykh (1, 2, 3, 4) , Delia Ionescu-Kruse (5)

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 UGA [2016-2019] - Université Grenoble Alpes [2016-2019]
5 IMAR - "Simion Stoilow" Institute of Mathematics

In the present study we consider three two-component (integrable and non-integrable) systems which describe the propagation of shallow water waves on a constant shear current. Namely, we consider the two-component Camassa-Holm equations, the Zakharov-Ito system and the Kaup--Boussinesq equations all including constant vorticity effects. We analyze both solitary and periodic-type travelling waves using the simple and geometrically intuitive phase space analysis. We get the pulse-type solitary wave solutions and the front solitary wave solutions. For the Zakharov-Ito system we underline the occurrence of the pulse and anti-pulse solutions. The front wave solutions decay algebraically in the far field. For the Kaup-Boussinesq system, interesting analytical multi-pulsed travelling wave solutions are found.

Mots clés: multi-pulsed solutions; phase-plane analysis; analytical solutions; Zakharov-Ito system; Kaup-Boussinesq equations; Camassa-Holm equations; solitary waves; cnoidal waves; waves on shear flow

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

Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete and Continuous Dynamical Systems - Series A, 2019, 39 (9), pp.5521-5541. ⟨10.3934/dcds.2019225⟩. ⟨hal-02094922⟩