The Improved Riccati Equation Method and Exact Solutions to mZK Equation

Abstract

We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

Exact solutions for the nonlinear KPP equation by using the Riccati equation method combined with the G/G - expansion method

“…In the following, we give the main steps of the Riccati equation method combined with the ( / ) GG  -expansion method (Li, 2012):…”

“…Many effective methods to construct the exact solutions of these equations have been established, such as, the inverse scattering transform method (Ablowitz and Clarkson, 1991), the Hirota method (Hirota, 1971), the truncated expansion method ( Weiss et al, 1983), the Backlund transform method (Miura, 1979;Rogers and Shadwick, 1982), the exp-function method (He and Wu, 2006;Yusufoglu, 2008), the tanh-function method (Fan, 2000;Zhang and Xia, 2008), the Jacobi elliptic function method (Chen and Wang, 2005;Lu, 2005), the ( ) GG  -expansion method (Wang and Zhang, 2008;Feng and Wan, 2011;Zayed ,*Corresponding author. E-mails: e.m.e.zayed@hotmail.com, yaser31270@yahoo.com Zayed and Abdelaziz, 2010;Zayed and El-Malky, 2011), the modified simple equation (Jawad,et al .,2010;Zayed, 2011;Zayed andHoda Ibrahim, 2012, Zayed andHoda Ibrahim, 2014;Zayed and Arnous, 2012), the Riccati equation method (Zhu, 2008;Li and Zhang, 2010;Zayed and Arnous, 2013), the improved Riccati equation method ( Li, 2012), the method of averaging (Leilei et al, 2014) and so on. The objective of this paper is to apply the improved Riccati equation method combined with the improved ( / ) GG  -expansion method to find the exact solutions of the following nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation:…”