Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations

Abstract

The KdV-KdV system of Boussinesq equations belongs to the class of Boussinesq equations modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid. It has been numerically shown that this system possesses solutions with two humps which tend to a periodic solution with much smaller amplitude at infinity (called generalized two-hump wave solutions). This paper presents the first rigorous proof. The traveling form of this system can be formulated into a dynamical system with dimension 4. The classical dynamical system approach provides the existence of a solution with an exponentially decaying part and an oscillatory part (small-amplitude periodic solution) at positive infinity, which has a single hump at the origin and is reversible near negative infinity if some free constants, such as the amplitude and the phase shit of the periodic solution, are activated. This eventually yields a generalized two-hump wave solution. The method here can be applied to obtain generalized $2^k$-hump wave solutions for any positive integer $k$.