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. In this work, 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 identified. An analytical solution of such peakons is derived also for a perturbed version of the compact equation. 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. 2012. ⟨hal-00687325v1⟩