  We discuss the application of a variant of the method of simplest equation
for obtaining exact traveling wave solutions of a class of nonlinear partial
differential equations containing polynomial nonlinearities. As simplest
equation we use differential equation for a special function that contains as
particular cases trigonometric and hyperbolic functions as well as the elliptic
function of Weierstrass and Jacobi. We show that for this case the studied
class of nonlinear partial differential equations can be reduced to a system of
two equations containing polynomials of the unknown functions. This system may
be further reduced to a system of nonlinear algebraic equations for the
parameters of the solved equation and parameters of the solution. Any
nontrivial solution of the last system leads to a traveling wave solution of
the solved nonlinear partial differential equation. The methodology is
illustrated by obtaining solitary wave solutions for the generalized
Korteweg-deVries equation and by obtaining solutions of the higher order
Korteweg-deVries equation.
