The extended Riccati equation mapping method for variable-coefficient diffusion–reaction and mKdV equations Shimin Guo a,b , Liquan Mei a,⇑ , Yubin Zhou b , Chao Li b a School of Science, Xi’an Jiaotong University, Xi’an, 710049, China b School of Mathematics and Statistics, Lan-zhou University, Lan-zhou 730000, China article info Keywords: Extended Riccati equation mapping method Nonlinear evolution equations Variable-coefficient diffusion–reaction equation Variable-coefficient mKdV equation abstract In this paper, the extended Riccati equation mapping method is proposed to seek exact solutions of variable-coefficient nonlinear evolution equations. Being concise and straight- forward, this method is applied to certain type of variable-coefficient diffusion–reaction equation and variable-coefficient mKdV equation. By means of this method, hyperbolic function solutions and trigonometric function solutions are obtained with the aid of sym- bolic computation. It is shown that the proposed method is effective, direct and can be used for many other variable-coefficient nonlinear evolution equations. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction By now, more and more nonlinear evolution equations (NLEEs) have arisen in many fields, such as fluid mechanics, plas- ma physics, plasma waves, hydrodynamic and so on. Searching for explicit solutions of NLEEs via various methods is quite important and interesting. In the past several years, many powerful methods for obtaining explicit solutions of NLEEs have been established and developed, such as the inverse scattering method [1], homogenous balance method [2–4], F-expansion method [5–7], sine–cosine method [8,9], tanh function method [10], sub-ODE method [11–13] and so on. However, most of the above methods are related to the constant-coefficient NLEEs. Recently, the study of the variable- coefficient NLEEs (vc-NLEEs) has become more and more attractive. That is because of the fact that a large number of impor- tant physical phenomena can be described by these equations. The present work is motivated by the desire to establish the extended Riccati equation mapping method to seek exact solutions of vc-NLEEs. As applications of the extended method, we will consider the following two vc-NLEEs. (i) Variable-coefficient diffusion–reaction (vc-DR equation) [16] u t ðx; tÞþ v ðtÞu x ðx; tÞ¼ Du xx ðx; tÞþ luðx; tÞ su 3 ðx; tÞ; ð1Þ where D is the diffusion coefficient, v is an arbitrary function of t, l and s are real constants. Thanks to the efforts of many researchers, certain types of nonlinear variable-coefficient diffusion–reaction equations have been investigated and solved [16–18]. (ii) Variable-coefficient mKdV equation (vc-mKdV equation) [14] u t ¼ K 0 ðtÞðu xxx  6u 2 u x Þþ 4K 1 ðtÞu x  hðtÞðu þ xu x Þ; ð2Þ where K 0 (t), K 1 (t) and h(t) are arbitrary functions of t. Eq. (2) is quite important in the mathematical physics fields, and mKdV equation and cylindrical mKdV equation are its special cases [15]. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.12.116 ⇑ Corresponding author. E-mail address: lqmei@mail.xjtu.edu.cn (L. Mei). Applied Mathematics and Computation 217 (2011) 6264–6272 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc