The partial differential equations of the form

${u}_{t}-\eta \mathrm{\Delta}{u}_{t}-\alpha \mathrm{\Delta}u=f(x,u,\mathrm{\nabla}u)$

(1)

arise in many areas of mathematics and physics, where $u=u(x,t)$, $x\in \mathrm{\Omega}\subset {\mathbb{R}}^{n}$, $t\ge 0$, *η* and *α* are non-negative constants, Δ denotes the Laplace operator acting on the space variables *x*. Equations of type (1) with only one time derivative appearing in the highest-order term are called pseudo-parabolic and they are a special case of Sobolev equations. They are characterized by derivatives of mixed type (*i.e.*, time and space derivatives together) appearing in the highest-order terms of the equation and were studied by Sobolev [1]. Sobolev equations have been used to describe many physical phenomena [2–8]. Equation (1) arises as a mathematical model for the unidirectional propagation of nonlinear, dispersive, long waves. In applications, *u* is typically the amplitude or velocity, *x* is proportional to the distance in the direction of propagation, and *t* is proportional to elapsed time [9].

An important special case of (1) is the Benjamin-Bona-Mahony-Burgers (BBMB) equation

$-{u}_{xxt}+{u}_{t}-\alpha {u}_{xx}+(1+u){u}_{x}=0.$

(2)

It has been proposed in [

10] as a model to study the unidirectional long waves of small amplitudes in water, which is an alternative to the Korteweg-de Vries equation of the form

${u}_{xxx}+{u}_{t}-{u}_{xx}+u{u}_{x}=0.$

(3)

The BBMB equation has been tackled and investigated by many authors. For more details, we refer the reader to [11–15] and the references therein.

In [

16], a generalized Benjamin-Bona-Mahony-Burgers equation

$-{u}_{xxt}+{u}_{t}-\alpha {u}_{xx}+\beta {u}_{x}+{(g(u))}_{x}=0$

(4)

has been considered and a set of new solitons, kinks, antikinks, compactons, and Wadati solitons have been derived using by the classical Lie method, where *α* is a positive constant, $\beta \in \mathbb{R}$, and $g(u)$ is a ${C}^{2}$-smooth nonlinear function. Equation (4) with the dissipative term $\alpha {u}_{xx}$ arises in the phenomena for both the bore propagation and the water waves.

Peregrine [17] and Benjamin, Bona, and Mahony [10] have proposed equation (4) with the parameters $g(u)=u{u}_{x}$, $\alpha =0$, and $\beta =1$. Furthermore, Benjamin, Bona, and Mahony proposed equation (4) as an alternative regularized long-wave equation with the same parameters.

Khaled, Momani, and Alawneh obtained explicit and numerical solutions of BBMB equation (4) by using the Adomian’s decomposition method [18] .

Tari and Ganji implemented variational iteration and homotopy perturbation methods obtaining approximate explicit solutions for (4) with $g(u)=\frac{{u}^{2}}{2}$ [19] and El-Wakil, Abdou, and Hendi used another method (the exp-function) to obtain the generalized solitary solutions and periodic solutions of this equation [20].

In addition, we consider $g(u)=\frac{{u}^{3}}{3}$ and obtain analytic solutions in a closed form.

The aim of this work is twofold. First, it is to obtain the exact solutions of the Benjamin-Bona-Mahony-Burgers (BBMB) equation and the generalized Benjamin-Bona-Mahony-Burgers equation with $g(u)=u{u}_{x}$, $g(u)=\frac{{u}^{2}}{2}$, $g(u)=\frac{{u}^{3}}{3}$; and second, it is to show that the tanh method can be applied to obtain the solutions of pseudo-parabolic equations.