A two-dimensional Boussinesq equation for water waves and some of its solutions

A two-dimensional Boussinesq equation, utt-uxx + 3(u2)xx - uxxxx - uyy = 0, is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation. A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied ; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta.