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

作者：Vitanov Nikolay K^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2011, 16(3): 1176-1185.

DOI：10.1016/j.cnsns.2010.06.011

• Nonlinear partial differential equations
• Modified method of simplest equation
• Swift-Hohenberg equation
• Generalized Rayleigh equation
• Traveling-wave solutions
• Equation of Bernoulli
• Equation of Riccati
• Extended tanh-equation

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.

• 出版日期2011-3