Solitary wave solutions for nonlinear partial differential equations that contain monomials of odd and even grades with respect to participating derivatives

•The method of simplest equation is applied to nonlinear PDEs.•The PDEs contain monomials of odd and even order with respect to derivatives.•Solitary wave solutions are studied.•Numerous examples of the methodology are presented.

We apply the method of simplest equation for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. We consider first the general case of presence of monomials of the both (odd and even) grades and then turn to the two particular cases of nonlinear equations that contain only monomials of odd grade or only monomials of even grade. The methodology is illustrated by numerous examples.