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

TL;DR

This paper explores special traveling wave solutions of a compact form of the Zakharov equation for deep-water gravity waves, including numerical construction, stability analysis, and the discovery of peakons and smooth solitary wave interactions.

Contribution

It introduces numerical methods to construct and analyze traveling wave solutions, including unstable peakons and evidence for integrability of the Zakharov equation.

Findings

Numerical construction of traveling wave solutions including peakons.

Unstable peakons with wedge-type singularities are discovered.

Smooth solitary waves appear to collide elastically, indicating integrability.

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.