General rogue waves in the Boussinesq equation.

Abstract

We derive general rogue wave solutions of arbitrary orders in the Boussinesq equation by the bilinear Kadomtsev-Petviashvili (KP) reduction method. These rogue solutions are given as Gram determinants with $2N-2$ free irreducible real parameters, where $N$ is the order of the rogue wave. Tuning these free parameters, rogue waves of various patterns are obtained, many of which have not been seen before. Compared to rogue waves in other integrable equations, a new feature of rogue waves in the Boussinesq equation is that the rogue wave of maximum amplitude at each order is generally asymmetric in space. On the technical aspect, our contribution to the bilinear KP-reduction method for rogue waves is a new judicious choice of differential operators in the procedure, which drastically simplifies the dimension reduction calculation as well as the analytical expressions of rogue wave solutions.