General rogue waves, breathers and hybrid structures of the coupled Boussinesq system

Abstract

In this paper, we concentrate on the rogue waves, breathers and hybrid solutions of the coupled Boussinesq system via the Kadomtsev–Petviashvili (KP) hierarchy reduction method. We construct the Gram determinant solutions for a $(2+1)$-dimensional bilinear system in the KP hierarchy which can be reduced to the coupled Boussinesq system. By considering the dimension-reduction condition, the general high-order rogue wave solutions expressed by derivatives with respect to parameters $p$ and $q$ are derived. For simplicity, the expressions of the rogue waves are replaced by purely algebraic ones with the help of the known Schur polynomials. The rogue waves from first till fourth order and their dynamic properties are numerically investigated. The structures of the $N$th-order rogue waves contain $\frac{N(N+1)}{2}$ first-order rogue waves. As more free parameters appear, the number of patterns increases. The breather solutions are obtained through setting specific parameter conditions in soliton solutions. Then first- and second-order breathers are attained and their dynamics are analyzed numerically. Three different arrangements for the first-order breathers as well as three types of second-order breather waves including interacting waves, parallel waves and coincident waves are displayed. The hybrid solutions containing first-order breather as well as first- and second-order solitons are given with dynamic analysis. A similar way can be used to obtain the $N$th-order rogue waves and the $M$th-order breathers. The method used in the paper can be extended to other integrable equations theoretically.