The tanh and the sine–cosine methods for exact solutions of the MBBM and the Vakhnenko equations

In this paper, we establish exact solutions for nonlinear evolution equations. The tanh and sine–cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations. Many new families of exact travelling wave solutions of the Vakhnenko and modified Benjamin–Bona–Mahony (MBBM) equations are successfully obtained. The obtained solutions include solitons, solitary and periodic solutions. These solutions may be important of significance for the explanation of some practical physical problems.