Numerical study of the generalised Klein-Gordon equations

Denys Dutykh (1, 2) , Marx Chhay (3) , Didier Clamond (4)

1 School of Mathematical Sciences [Dublin]
2 LAMA - Laboratoire de Mathématiques
3 LOCIE - LabOratoire proCédés énergIe bâtimEnt
4 JAD - Laboratoire Jean Alexandre Dieudonné

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalized Klein-Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets.

Mots clés: deep water approximation; spectral methods; travelling waves; periodic waves; stability

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Physique Générale [physics.gen-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Atmosphérique et Océanique [physics.ao-ph]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Analyse classique [math.CA]; Physique Numérique [physics.comp-ph]

Denys Dutykh, Marx Chhay, Didier Clamond. Numerical study of the generalised Klein-Gordon equations. 2013. ⟨hal-00851030v1⟩