Explicit series solution of travelling waves with a front of Fisher equation Yue Tan, Hang Xu, Shi-Jun Liao * School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, China Accepted 3 October 2005 Communicated by L. Reichl Abstract In this paper, an analytic technique, namely the homotopy analysis method, is employed to solve the Fisher equa- tion, which describes a family of travelling waves with a front. The explicit series solution for all possible wave speeds 0< c <+1 is given. Such kind of explicit series solution has never been reported, to the best of authorÕs knowledge. Our series solution indicates that the solution contains an oscillation part when 0 < c < 2. The proposed analytic approach is general, and can be applied to solve other similar nonlinear travelling wave problems. Ó 2005 Elsevier Ltd. All rights reserved. 1. Introduction The nonlinear reaction–diffusion equation ou ot ¼ D o 2 u ox 2 þ muð1  uÞ; ð1Þ was first introduced by Fisher as a model for the propagation of a mutant gene [1]. It has wide application in the fields of logistic population growth [2,3], flame propagation [4], neurophysiology [5], autocatalytic chemical reactions [6], branching Brownian motion processes [7], and nuclear reactor theory [8]. In chemical media the function u(x, t) is the concentration of the reactant. D represents its diffusion coefficient, and the positive constant m specifies the rate of chemical reaction. In media of other natures, u might be temperature or elec- tric potential, D might be the thermal conductivity or specific electrical conductivity. The medium described by Eq. (1) is often referred to as a bistable medium because it has two homogeneous stationary states, u = 0 and u = 1. A kink-like travelling wave solution of Eq. (1) describes a constant-velocity front of transition from one homogeneous state to another. 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.001 * Corresponding author. Tel.: +86 21 6293 2676; fax: +86 21 6293 3156. E-mail address: sjliao@sjtu.edu.cn (S.-J. Liao). Chaos, Solitons and Fractals 31 (2007) 462–472 www.elsevier.com/locate/chaos