The Traveling Wave Solutions and Their Bifurcations for the BBM-Like B(m,n) Equations

Abstract

We investigate the traveling wave solutions and their bifurcations for the BBM-like $B(m,n)$ equations $u_t+\alpha u_x+\beta {(u^m)}_x-\gamma {(u^n)}_{xxt}=0$ by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like $B(3,2)$ equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like $B(4,2)$ equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.