Special solutions to a compact equation for deep-water gravity waves

Francesco Fedele (1, 2) , Denys Dutykh (3)

1 School of Civil and Environmental Engineering [Atlanta]
2 ECE GeorgiaTech - School of Electrical and Computer Engineering - Georgia Insitute of Technology
3 LAMA - Laboratoire de Mathématiques

Recently, Dyachenko & Zakharov (2011) have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. Further, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of singular traveling waves, we consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fourier-type spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.

Mots clés: water waves; deep water approximation; Hamiltonian structure; travelling waves; solitons

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

Francesco Fedele, Denys Dutykh. Special solutions to a compact equation for deep-water gravity waves. Journal of Fluid Mechanics, 2012, 712, pp.646-660. ⟨10.1017/jfm.2012.447⟩. ⟨hal-00687325v3⟩