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

We discuss the class of equations ∑ i , j = 0 m A ij ( u ) ∂ i u ∂ t i ∂ u ∂ t j + ∑ k , l = 0 n B kl ( u ) ∂ k u ∂ x k ∂ u ∂ x l = 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.