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

Abstract

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.

17 pages, 14 figures, 41 references. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/

References in corpus (3)

• A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
• Hamiltonian form and solitary waves of the spatial Dysthe equations
• Collision of two breathers at surface of deep water

Cited by in corpus (9)

• Efficient computation of steady solitary gravity waves
• On certain properties of the compact Zakharov equation
• Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering
• Collision of two breathers at surface of deep water
• Crest speeds of unsteady surface water waves
• High-order nonlinear Schrödinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions
• Vortexons in axisymmetric Poiseuille pipe flows
• Camassa-Holm equations and vortexons for axisymmetric pipe flows
• Soliton Collision in Random Seas