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

Dyachenko & Zakharov (J. Expl Theor. Phys. Lett., vol. 93, 2011, pp. 701-705) recently 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 travelling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. In particular, unstable travelling waves with wedge-type singularities, namely peakons, are numerically discovered. To gain insight into the properties of these singular solutions, we also 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.