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

摘要

We discuss the class of equations m∑ij=0 A_(ij)(u)((δ)~iu)/((δ)t~i)((δ)u)/((δ)t)+n∑k,l=0 B_(kl)(u)((δ)u)/((δ)x)~1=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.