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We discuss the application of a variant of the method of simplest equationfor obtaining exact traveling wave solutions of a class of nonlinear partialdifferential equations containing polynomial nonlinearities. As simplestequation we use differential equation for a special function that contains asparticular cases trigonometric and hyperbolic functions as well as the ellipticfunction of Weierstrass and Jacobi. We show that for this case the studiedclass of nonlinear partial differential equations can be reduced to a system oftwo equations containing polynomials of the unknown functions. This system maybe further reduced to a system of nonlinear algebraic equations for theparameters of the solved equation and parameters of the solution. Anynontrivial solution of the last system leads to a traveling wave solution ofthe solved nonlinear partial differential equation. The methodology isillustrated by obtaining solitary wave solutions for the generalizedKorteweg-deVries equation and by obtaining solutions of the higher orderKorteweg-deVries equation.

Modified method of simplest equation for obtaining exact analytical  solutions of nonlinear partial differential equations: Further development of  methodology with two applications