Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs

We discuss the class of equations Sigma(m)(ij=0)A(ij)(u) partial derivative(i)u/partial derivative t(i) (partial derivative u/partial derivative t)(j) + Sigma(n)(k,l=0) B-kl(u) partial derivative(k)u/partial derivative x(k) (partial derivative u/partial derivative x)(t) = C(u) where A(ij)(u), B-kl(u) and C(u) are functions of u(x, t) as follows: (i) A(ij), B-kl and C are polynomials of u; or (ii) A(ij), B-kl, and C can be reduced to polynomials of u by means of Taylor series for small values of u. For these two cases the above-mentioned class of equations consists of nonlinear PDEs with polynomial nonlinearities. We show that the modified method of simplest equation is powerful tool for obtaining exact traveling-wave solution of this class of equations. The balance equations for the sub-class of traveling-wave solutions of the investigated class of equations are obtained. We illustrate the method by obtaining exact traveling-wave solutions (i) of the Swift-Hohenberg equation and (ii) of the generalized Rayleigh equation for the cases when the extended tanh-equation or the equations of Bernoulli and Riccati are used as simplest equations. (C) 2010 Elsevier B.V. All rights reserved.