%0 Journal Article
%T The General Traveling Wave Solutions of the Fisher Equation with Degree Three
%A Wenjun Yuan
%A Qiuhui Chen
%A Jianming Qi
%A Yezhou Li
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/657918
%X We employ the complex method to research the integrality of the Fisher equations with degree three. We obtain the sufficient and necessary condition of the integrable of the Fisher equations with degree three and the general meromorphic solutions of the integrable Fisher equations with degree three, which improves the corresponding results obtained by Feng and Li (2006), Guo and Chen (1991), and A£¿£¿rseven and £¿zi£¿ (2010). Moreover, all are new general meromorphic solutions of the Fisher equations with degree three for . Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics. 1. Introduction and Main Result Consider the Fisher equation which is a nonlinear diffusion equation as a model for the propagation of a mutant gene with an advantageous selection intensity . It was suggested by Fisher as a deterministic version of a stochastic model for the spatial spread of a favored gene in a population in 1936. Set and and drop the primes; (1) becomes Substituting the traveling wave transform , into (2), it gives a nonlinear ordinary differential equation where is a constant. Finding solutions of nonlinear models is a difficult and challenging task. In 2005 and 2009, Feng et al. [1, 2] proposed an analytic method to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation were obtained explicitly. These solutions included solitary wave solutions, singular traveling wave solutions, and periodical wave solutions. These results indicated that in some cases their analytic approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems. In 2010 and 2011, Demina et al. [3¨C5] studied the meromorphic solutions of autonomous nonlinear ordinary differential equations. An algorithm for constructing meromorphic solutions in explicit form was presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) were found for a wide class of autonomous nonlinear ordinary differential equations. Recently, the complex method was introduced by Yuan et al. [6¨C8]. Very recently, Yuan et al. [9] employed the complex method to obtain the general solutions of (3). In order to state our results, we need some concepts and notations. A meromorphic function means that is
%U http://www.hindawi.com/journals/amp/2013/657918/