On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis–Processi equation and b-equation

Abstract

The method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. Here we extend the class of equations which can be treated by the method in such a way that the classes of equations considered in our previous work are particular cases of the extended class of equations. As examples for application of the methodology we obtain exact traveling-wave solutions of the generalized Degasperis–Processi equation and of the b-equation. As simplest equations we use the equations of Bernoulli and Riccati. We investigate the possibility for obtaining these solutions also by means of the exp-function method. This lead us to propose a generalized version of the exp-function method in Section 5.