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