Some Further Results on Traveling Wave Solutions for the ZK-BBM(m,n) Equations

Abstract

We investigate the traveling wave solutions for the ZK-BBM($m,n$) equations $u_t+u_x-a{\left(u^m\right)}_x+{\left(b{\left(u^n\right)}_{xt}+k{\left(u^n\right)}_{yt}\right)}_x=0$ by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.