On the exact solitary wave solutions of a special class of Benjamin–Bona–Mahony equation

The general form of Benjamin-Bona-Mahony equation (BBM) is $ u_t+au_x+bu_{xxt}+(g(u))_x=0,\quad a,b\in\mathbb{R}$, where $ab\ne0$ and $g(u)$ is a $C^2$-smooth nonlinear function, has been proposed by Benjamin et al. In [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin–Bona–Mahony equation (BBM) $u_t+au_x+bu_{xxt}+(pe^u+qe^{-u})_x=0$, $a, b, p, q \in\mathbb{R}$, where $ab\ne0$, and $qp\ne0$, and we obtain new exact solutions for it by using the well-known $(G'/G)$-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.