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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 489043, 11 pages
http://dx.doi.org/10.1155/2012/489043
Research Article

The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation

1School of Computer Science and Educational Software, Guangzhou University, Guangzhou 510006, China
2School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received 25 October 2011; Accepted 14 November 2011

Academic Editor: Shaher M. Momani

Copyright © 2012 Yong Huang and Yadong Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Bäcklund transformations and abundant exact explicit solutions for a class of nonlinear wave equation are obtained by the extended homogeneous balance method. These solutions include the solitary wave solution of rational function, the solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. In addition to rederiving some known solutions, some entirely new exact solutions are also established. Explicit and exact particular solutions of many well-known nonlinear evolution equations which are of important physical significance, such as Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system, are obtained as special cases.

1. Introduction

The existence of solitary wave solutions and periodic wave solutions is an important question in the study of nonlinear evolution equations. The methods of finding such solutions for integrable equations are well known: the solitary wave solutions can be found by inverse scattering transformation [1] and the Hirota bilinear method [2], and the periodic solutions can be represented by sums of equally spaced solitons represented by sech-function [3, 4]. Weiss et al. developed the singular manifold method to introduce the Painlevé property in the theory of partial differential equations [5]. The beauty of the singular manifold method is that this expansion for a nonlinear PDE contains a lot of information about this PDE. For an equation that possesses the Painlevé property the singular manifold method leads to the Bäcklund transformation, the Lax pair, and Miura transformations and makes connections to the Hirota bilinear method, Laplace-Darboux transformations [6]. Most nonlinear nonintegrable equations do not possess the Painlevé property; that is, they are not free from “movable” critical singularities. For some nonintegrable nonlinear equations it is still possible to obtain single-value expansions by putting a constraint on the arbitrary function in the Painlevé expansion. Such equations are said to be partially integrable, and Weiss [7] conjectured that these systems can be reduced to integrable equations. Another treatment of the partially integrable systems was offered by Hietarinta [8] by the generalization of the Hirota bilinear formalism for nonintegrable systems. He conjectured that all completely integrable PDEs can be put into a bilinear form. There are also nonintegrable equations that can be put into the bilinear form and then the partial integrability is associated with the levels of integrability defined by the number of solitons that can be combined to an N-soliton solution. Partial integrability then means that the equation allows a restricted number of multisoliton solutions. In [9] Berloff and Howard suggested joining these treatments of the partial nonintegrability and using the Painlevé expansion truncated before the “constant term” level as the transform for reducing a nonintegrable PDE to a multilinear equation.

The Bäcklund transformation is not only a useful tool to obtain exact solutions of some soliton equation from a trivial “seed” but also related to infinite conservation laws and inverse scattering method [1]. In [1012], Wang Mingliang proposed the homogeneous balance method—an effective method solving nonlinear partial differential equations. Fan and Zhang extended the homogeneous balance method and proposed an approach to obtain Bäcklund transformation for the nonlinear evolution equations [13]. In a recent paper [14], Shang obtained the Bäcklund transformation, a Lax pair, and some new explicit exact solutions of Hirota-Satsuma SWW equation (2.3) by means of the Bäcklund transformations and the extension of the hyperbolic function method presented in [15].

In this paper we investigate a general nonintegrable nonlinear convection-diffusion equation𝑢𝑡𝑢𝑥𝑥+𝛼𝑢𝑢𝑥+𝛽𝑢+𝛾𝑢2+𝛿𝑢3=0,(1.1) where 𝛼, 𝛽, 𝛾, and 𝛿 are arbitrary real constants. Equation (1.1) include many well-known nonlinear equations that are with applied background as special examples, such as Burgers equation, Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction-diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system. The extended homogeneous balance method is applied for a reliable treatment of the nonintegrable nonlinear equation (1.1). Some Bäcklund transformations and abundant explicit exact particular solutions of the nonintegrable nonlinear equation (1.1) are obtained by means of the extended homogeneous balance method. Some explicit exact solutions obtained here have more general form than some known solutions, and some explicit exact solutions obtained here are entirely new solutions.

2. Bäcklund Transformations for the Nonintegrable Nonlinear Wave Equation

According to the extended homogeneous balance method, we suppose that the solution of (1.1) is of the form𝑢(𝑥,𝑡)=𝑓(𝜙)𝜙𝑥+𝑢1(𝑥,𝑡),(2.1) where 𝑓, 𝜙 are two functions to be determined and 𝑢1(𝑥,𝑡) is a solution of (1.1).

From (2.1), we have𝑢𝑡=𝑓(𝜙)𝜙𝑥𝜙𝑡+𝑓(𝜙)𝜙𝑥𝑡+𝑢1𝑡𝑢,(2.2)𝑥=𝑓(𝜙)𝜙2𝑥+𝑓(𝜙)𝜙𝑥𝑥+𝑢1𝑥𝑢,(2.3)𝑥𝑥=𝑓(𝜙)𝜙3𝑥+3𝑓(𝜙)𝜙𝑥𝜙𝑥𝑥+𝑓(𝜙)𝜙𝑥𝑥𝑥+𝑢1𝑥𝑥𝑢,(2.4)2=𝑓2(𝜙)𝜙2𝑥+2𝑓𝜙𝑥𝑢1(𝑥,𝑡)+𝑢21𝑢(𝑥,𝑡),3=𝑓3(𝜙)𝜙3𝑥𝑓+32𝜙2𝑥𝑢1(𝑥,𝑡)+3𝑓𝜙𝑥𝑢21(𝑥,𝑡)+𝑢31(𝑥,𝑡).(2.5) Substituting (2.1)–(2.5) into the left side of (1.1) and collecting all terms with 𝜙3𝑥, we obtain𝑢𝑡𝑢𝑥𝑥+𝛼𝑢𝑢𝑥+𝛽𝑢+𝛾𝑢2+𝛿𝑢3=𝛼𝑓𝑓𝑓𝑓+𝛿3𝜙3𝑥+𝑓𝜙𝑥𝜙𝑡3𝑓𝜙𝑥𝜙𝑥𝑥+𝛼𝑓𝜙2𝑥𝑢1𝑓+𝛼2𝜙𝑥𝜙𝑥𝑥𝑓+𝛾2𝜙2𝑥𝑓+3𝛿2𝜙2𝑥𝑢1(𝑥,𝑡)+𝑓𝜙𝑥𝑡𝜙𝑥𝑥𝑥+𝛼𝜙𝑥𝑥𝑢1+𝛼𝜙𝑥𝑢1𝑥+𝛽𝜙𝑥+2𝛾𝜙𝑥𝑢1+3𝛿𝜙𝑥𝑢21+𝑢1𝑡𝑢1𝑥𝑥+𝛼𝑢1𝑢1𝑥+𝛽𝑢1+𝛾𝑢21+𝛿𝑢31=0.(2.6) Setting the coefficient of 𝜙3𝑥 in (2.6) to be zero, we obtain an ordinary differential equation for 𝑓𝛼𝑓𝑓𝑓𝑓+𝛿3=0,(2.7) which has a solution𝑓(𝜙)=𝜆ln(𝜙),(2.8) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿. And then𝑓2=(𝜆)𝑓.(2.9) By virtue of (2.7)–(2.9), (2.6) becomes𝑢𝑡𝑢𝑥𝑥+𝛼𝑢𝑢𝑥+𝛽𝑢+𝛾𝑢2+𝛿𝑢3=𝑓𝜙𝑥𝜙𝑡3𝜙𝑥𝜙𝑥𝑥+𝛼𝜙2𝑥𝑢1𝛼𝜆𝜙𝑥𝜙𝑥𝑥𝛾𝜆𝜙2𝑥3𝛿𝜆𝜙2𝑥𝑢1(𝑥,𝑡)+𝑓𝜙𝑥𝑡𝜙𝑥𝑥𝑥+𝛼𝜙𝑥𝑥𝑢1+𝛼𝜙𝑥𝑢1𝑥+𝛽𝜙𝑥+2𝛾𝜙𝑥𝑢1+3𝛿𝜙𝑥𝑢21+𝑢1𝑡𝑢1𝑥𝑥+𝛼𝑢1𝑢1𝑥+𝛽𝑢1+𝛾𝑢21+𝛿𝑢31=0.(2.10) Setting the coefficients of 𝑓,𝑓,𝑓0 to be zero, respectively, it is easy to see from (2.10) that𝜙𝑡+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝜙𝑥(3+𝛼𝜆)𝜙𝑥𝑥𝜙=0,(2.11)𝑥𝑡𝜙𝑥𝑥𝑥+𝛼𝜙𝑥𝑥𝑢1+𝛼𝜙𝑥𝑢1𝑥+𝛽𝜙𝑥+2𝛾𝜙𝑥𝑢1+3𝛿𝜙𝑥𝑢21𝑢=0,(2.12)1𝑡𝑢1𝑥𝑥+𝛼𝑢1𝑢1𝑥+𝛽𝑢1+𝛾𝑢21+𝛿𝑢31=0.(2.13) Substituting (2.8) into (2.1), we obtain a Bäcklund transformation𝜙𝑢(𝑥,𝑡)=𝜆𝑥𝜙+𝑢1(𝑥,𝑡),(2.14) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿,𝜙,𝑢1 satisfy (2.11)–(2.13). Substituting a seed solution 𝑢1(𝑥,𝑡) of (1.1) into linear equations (2.11) and (2.12), then solving (2.11) and (2.12), we can get a new solution of (1.1) from (2.14). Thus we can obtain infinite solutions of (1.1) by the Bäcklund transformation (2.14) and (2.11)-(2.12) from a seed solution of (1.1).

Taking 𝑢1=0, by (2.11)–(2.14), we obtain a transformation𝜙𝑢(𝑥,𝑡)=𝜆𝑥𝜙,(2.15) that transforms (1.1) into linear equations𝜙𝑡𝛾𝜆𝜙𝑥(3+𝛼𝜆)𝜙𝑥𝑥𝜙=0,𝑡𝜙𝑥𝑥+𝛽𝜙=𝐸,(2.16) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝐸 is an arbitrary constant.

Taking 𝑢1=(𝛾±Δ)/2𝛿, from (2.11)–(2.14) we obtain another transformation𝑢(𝑥,𝑡)=𝛾±Δ𝜙2𝛿+𝜆𝑥𝜙.(2.17) Equation (1.1) can be solved by solving two linear equations 𝜙𝑡+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝜙𝑥(3+𝛼𝜆)𝜙𝑥𝑥𝜙=0,𝑥𝑡𝜙𝑥𝑥𝑥+𝛼𝜙𝑥𝑥+𝛽𝜙𝑥+2𝛾𝜙𝑥𝑢1+3𝛿𝜙𝑥𝑢21=0,(2.18) where 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿.

3. Exact Explicit Solutions to (1.1)

In this section we want to obtain abundant exact explicit particular solutions of (1.1) from the Bäcklund transformation (2.14) and a trivial solution of (1.1).

Noting the homogeneous property of (2.16) we can expect that 𝜙 in (2.16) is of the form𝜙(𝑥,𝑡)=𝐴sinh𝑘𝑥+𝜔𝑡+𝜉0+𝐵cosh𝑘𝑥+𝜔𝑡+𝜉0+𝐶(3.1) with 𝐴, 𝐵, 𝐶, 𝑘, 𝜔, and 𝜉0 constants to be determined. Substituting (3.1) into (2.16), one gets a set of nonlinear algebraic equation𝐴𝜔𝛾𝜆𝐴𝑘(3+𝛼𝜆)𝐵𝑘2=0,𝐵𝜔𝛾𝜆𝐵𝑘(3+𝛼𝜆)𝐴𝑘2=0,𝐴𝜔𝐵𝑘2+𝛽𝐵=0,𝐵𝜔𝐴𝑘2+𝛽𝐴=0,𝛽𝐶=𝐸.(3.2) Solving (3.2), we have the following.

Case 1. 𝐴=𝐵, 𝐶=𝐸/𝛽, 𝜔=𝑘2𝛽, and 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+𝛾𝜆𝑘+𝛽=0.

Case 2. 𝐴=𝐵, 𝐶=𝐸/𝛽, 𝜔=𝛽𝑘2, and 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2𝛾𝜆𝑘+𝛽=0.

Thus we obtain the following explicit exact solutions of (1.1) given by𝑢(𝑥,𝑡)=𝜆𝑘exp𝑘𝑥+𝜔t+𝜉0exp𝑘𝑥+𝜔𝑡+𝜉0+𝐶,(3.3) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜔=𝑘2𝛽, 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+𝛾𝜆𝑘+𝛽=0, 𝐶0, and 𝜉0 are arbitrary constants.

We can also obtain the following explicit exact solutions of (1.1) given by1𝑢(𝑥,𝑡)=𝜆𝑘𝐶exp𝑘𝑥+𝜔𝑡+𝜉0,1(3.4) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜔=𝛽𝑘2, 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2𝛾𝜆𝑘+𝛽=0, 𝐶0, and 𝜉0 are arbitrary constants.

By direct computation, we readily obtain the following two useful formulas:exp(𝜉)=1𝐶+exp(𝜉)1,for𝐶=0,21tanh21(𝜉ln𝐶)+1,for𝐶>0,21coth21(𝜉ln(𝐶))+1,for𝐶<0,(3.5)=1𝐶exp(𝜉)11,for𝐶=0,21coth2(1𝜉+ln𝐶)1,for𝐶>0,21tanh2(𝜉+ln(𝐶))1,for𝐶<0,(3.6) where 𝐶 is arbitrary.

Thanks to the two formulas (3.5) and (3.6), we can assert.

The solutions (3.3) ((3.4), resp.) are soliton solutions of kink type in the case of 𝐶>0 (𝐶<0, resp.).

The solutions (3.3) ((3.4), resp.) are soliton-like solutions of singular type in the csae of 𝐶<0 (𝐶>0, resp.).

Analogously, we assume that 𝜙 in (2.16) is of the form𝜙(𝑥,𝑡)=𝐴sin𝑘𝑥+𝜔𝑡+𝜉0+𝐵cos𝑘𝑥+𝜔𝑡+𝜉0+𝐶(3.7) with 𝐴, 𝐵, 𝐶, 𝑘, 𝜔, and 𝜉0 constants to be determined. Substituting (3.7) into (2.16), one gets a set of nonlinear algebraic equation𝐴𝜔𝛾𝜆𝐴𝑘+(3+𝛼𝜆)𝐵𝑘2=0,𝐵𝜔+𝛾𝜆𝐵𝑘+(3+𝛼𝜆)𝐴𝑘2=0,𝐴𝜔+𝐵𝑘2+𝛽𝐵=0,𝐵𝜔+𝐴𝑘2+𝛽𝐴=0,𝛽𝐶=𝐸.(3.8) Solving (3.8), we have the following.

Case 1. 𝐴=𝐵𝑖, 𝐶=𝐸/𝛽, 𝜔=(𝑘2+𝛽)𝑖, and 𝑘 is a root of second order algebraic equation (2+𝛼𝜆)𝑘2𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

Case 2. 𝐴=𝐵𝑖, 𝐶=𝐸/𝛽, 𝜔=𝑖(𝑘2+𝛽), and 𝑘 is a root of second order algebraic equation (2+𝛼𝜆)𝑘2+𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

According to the result of Case 1, from (2.15) and (3.7), we obtain the exact explicit solutions of (1.1) given by𝑢(𝑥,𝑡)=𝜆𝑘𝑖exp(𝑖𝜉),exp(𝑖𝜉)+𝐶(3.9) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=𝑖(𝑘2+𝛽), 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

By the result of Case 2 and (2.15), (3.7), we can obtain the following exact explicit solutions of (1.1) given by1𝑢(𝑥,𝑡)=(𝜆𝑘𝑖),1+𝐶exp(𝑖𝜉)(3.10) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=(𝑖)(𝑘2+𝛽), 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

Analogously, we have the following two useful formulas:exp𝑖(𝜉)=1𝐶+exp𝑖(𝜉)1,for𝐶=0,21𝑖tan21(𝜉+𝑖ln𝐶)+1,for𝐶>0,21𝑖cot21(𝜉+𝑖ln(𝐶))+1,for𝐶<0,(3.11)=1𝐶exp𝑖(𝜉)+11,for𝐶=0,211𝑖tan2(1𝜉𝑖ln𝐶),for𝐶>0,211+𝑖cot2(𝜉𝑖ln(𝐶)),for𝐶<0.(3.12)

Due to the formula (3.11), we have from(3.9)𝑢(𝑥,𝑡)=𝜆𝑘21tan2𝑘𝑥+𝜔𝑡+𝜉0++𝑖ln(𝐶)𝜆𝑘𝑖2,for𝐶>0,𝜆𝑘21cot2𝑘𝑥+𝜔𝑡+𝜉0++𝑖ln(𝐶)𝜆𝑘𝑖2,for𝐶<0,(3.13) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=𝑖(𝑘2+𝛽), 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

Owing to the formula (3.12), we have from  (3.10)𝑢(𝑥,𝑡)=𝜆𝑘21tan2𝑘𝑥+𝜔𝑡+𝜉0𝑖ln(𝐶)𝜆𝑘𝑖2,for𝐶>0,𝜆𝑘21cot2𝑘𝑥+𝜔𝑡+𝜉0𝑖ln(𝐶)𝜆𝑘𝑖2,for𝐶<0,(3.14) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=(𝑖)(𝑘2+𝛽), 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+𝛾𝜆𝑘𝑖𝛽=0, 𝑖=1.

By virtue of the homogeneous property of (2.18), we can expect that 𝜙 is of the linear function form𝜙(𝑥,𝑡)=𝑘𝑥+𝜔𝑡+𝜉0,(3.15) with 𝑘 and 𝜔, 𝜉0 constants to be determined. Substituting (3.15) into (2.18), we find that (3.15) satisfies (2.18), provided that 𝑘 and 𝜔 satisfy the following algebraic equations:𝜔+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝑘=0,𝛽𝑘+2𝛾𝑘𝑢1+3𝛿𝑘𝑢21=0,(3.16) where 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿. Solving (3.16), we obtain that𝜔=𝛼𝛼2+8𝛿4𝛿𝛾𝑘,𝑘=arbitraryconstant,𝑢1𝛾=2𝛿,(3.17) provided that coefficients 𝛽, 𝛾, and 𝛿 of (1.1) satisfy condition 𝛾2=4𝛽𝛿.

Substituting (3.15) with (3.17) into (2.17), we obtain the exact particular solutions of  (1.1)𝛾𝑢(𝑥,𝑡)=+2𝛿𝛼±𝛼2+8𝛿12𝛿𝑥+𝛼𝛼2+8𝛿/4𝛿𝛾𝑡+𝜉0.(3.18)

Now we suppose that (2.18) has solutions of the form (3.1) substituting (3.1) into (2.18), one gets a set of algebraic equations: 𝐴𝜔+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝐴𝑘(3+𝛼𝜆)𝐵𝑘2=0,𝐵𝜔+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝐵𝑘(3+𝛼𝜆)𝐴𝑘2=0,𝐴𝑘𝜔𝐵𝑘3+𝛼𝐴𝑘2+𝛽𝐵𝑘+2𝛾𝑢1𝐵k+3𝛿𝑢21𝐵𝑘=0,𝐵𝑘𝜔𝐴𝑘3+𝛼𝐵𝑘2+𝛽𝐴𝑘+2𝛾𝑢1𝐴𝑘+3𝛿𝑢21𝐴𝑘=0.(3.19) In order to obtain nontrivial solutions of (1.1), we need to require that 𝑘, 𝜔 are all nonzero constants. Solving (3.19), one gets the following solutions.

Case 1. One has 𝐴=𝐵,𝐶=arbitraryconstant,𝜔=𝑘2𝛼𝑘𝛽2𝛾𝑢13𝛿𝑢21,or𝜔=(3+𝛼𝜆)𝑘2+𝛾𝜆+3𝛿𝜆𝑢1𝛼𝑢1𝑘,(3.20) where 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+(𝛾𝜆+3𝛿𝜆𝑢1+𝛼𝛼𝑢1)𝑘+𝛽+2𝛾𝑢1+3𝛿𝑢21=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿.

Case 2. One has 𝐴=𝐵,𝐶=arbitraryconstant,𝜔=𝛽+2𝛾𝑢1+3𝛿𝑢21𝑘2𝛼𝑘,or𝜔=𝛾𝜆+3𝛿𝜆𝑢1𝛼𝑢1𝑘(3+𝛼𝜆)𝑘2,(3.21) where 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+(𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝛼)𝑘+𝛽+2𝛾𝑢1+3𝛿𝑢21=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿.

By Case 1, we obtain the exact solutions of the (1.1) from (2.17),  (3.1)𝑢(𝑥,𝑡)=𝛾±Δ2𝛿+𝜆𝑘exp𝑘𝑥+𝜔𝑡+𝜉0exp𝑘𝑥+𝜔𝑡+𝜉0,+𝐶(3.22) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿, 𝜔=𝑘2𝛼𝑘𝛽2𝛾𝑢13𝛿𝑢21, 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+(𝛾𝜆+3𝛿𝜆𝑢1+𝛼𝛼𝑢1)𝑘+𝛽+2𝛾𝑢1+3𝛿𝑢21=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜉0, 𝐶0 are arbitrary constants.

According to the result of Case 2 and (2.17), (3.1), one obtain the other exact solutions𝑢(𝑥,𝑡)=𝛾±Δ2𝛿+𝜆𝑘exp𝑘𝑥+𝜔𝑡+𝜉0𝐶exp𝑘𝑥+𝜔𝑡+𝜉0,1(3.23) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿, 𝜔=𝛽+2𝛾𝑢1+3𝛿𝑢21𝑘2𝛼𝑘, 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑘2+(𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝛼)𝑘+𝛽+2𝛾𝑢1+3𝛿𝑢21=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜉0, 𝐶0 are arbitrary constants.

According to formulas (3.5), (3.6), we can get multiple new soliton solutions of kink type and multiple new soliton-like solutions of singular type from (3.22) and (3.23).

Analogously, we assume that (2.18) has solutions of the form (3.7); substituting (3.7) into (2.18), one gets a set of algebraic equations𝐴𝜔+𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝐴𝑘+(3+𝛼𝜆)𝐵𝑘2=0,𝐵𝜔𝛼𝑢1𝛾𝜆3𝛿𝜆𝑢1𝐵𝑘+(3+𝛼𝜆)𝐴𝑘2=0,𝐴𝑘𝜔𝐵𝑘3𝛼𝐴𝑘2𝛽𝐵𝑘2𝛾𝑢1𝐵𝑘3𝛿𝑢21𝐵𝑘=0,𝐵𝑘𝜔+𝐴𝑘3𝛼𝐵𝑘2+𝛽𝐴𝑘+2𝛾𝑢1𝐴𝑘+3𝛿𝑢21𝐴𝑘=0.(3.24) In order to obtain a nontrivial solution of (1.1), we also need to assume that 𝑘,𝜔 are all nonzero constants. Solving (3.24), we obtain the following.

Case 1. One has 𝐴=𝐵𝑖,𝐶=arbitraryconstant,𝜔=𝑖𝑘2𝛼𝑘+𝑖𝛽+2𝛾𝑢1+3𝛿𝑢21,or𝜔=(3+𝛼𝜆)𝑖𝑘2+𝛾𝜆+3𝛿𝜆𝑢1𝛼𝑢1𝑘,(3.25) where 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑖𝑘2+(𝛾𝜆+3𝛿𝜆𝑢1+𝛼𝛼𝑢1)𝑘𝑖(𝛽+2𝛾𝑢1+3𝛿𝑢21)=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿.

Case 2. One has 𝐴=𝐵𝑖,𝐶=arbitraryconstant,𝜔=𝑖𝑘2𝛼𝑘𝑖𝛽+2𝛾𝑢1+3𝛿𝑢21,or𝜔=(3+𝛼𝜆)𝑖𝑘2+𝛾𝜆+3𝛿𝜆𝑢1𝛼𝑢1𝑘,(3.26) where 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑖𝑘2(𝛾𝜆+3𝛿𝜆𝑢1+𝛼𝛼𝑢1)𝑘𝑖(𝛽+2𝛾𝑢1+3𝛿𝑢21)=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿.

Collecting (2.17), (3.7), (3.25), and (3.26), we obtain the following explicit exact periodic traveling wave solutions𝑢(𝑥,𝑡)=𝛾±Δ2𝛿+𝑖𝜆𝑘exp(𝑖𝜉)exp(𝑖𝜉)+𝐶,(3.27) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=𝑖𝑘2𝛼𝑘+𝑖(𝛽+2𝛾𝑢1+3𝛿𝑢21), 𝑘 is a root of second-order algebraic equation (2+𝛼𝜆)𝑖𝑘2 + (𝛾𝜆+3𝛿𝜆𝑢1 + 𝛼𝛼𝑢1)𝑘𝑖(𝛽+2𝛾𝑢1 + 3𝛿𝑢21)=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜉0, 𝐶0 are arbitrary constants,𝑢(𝑥,𝑡)=𝛾±Δ2𝛿𝑖𝜆𝑘𝐶exp(𝑖𝜉),exp(𝑖𝜉)+1(3.28) where 𝜆=(𝛼±𝛼2+8𝛿)/2𝛿, Δ=𝛾24𝛽𝛿, 𝜉=𝑘𝑥+𝜔𝑡+𝜉0, 𝜔=𝑖𝑘2𝛼𝑘𝑖(𝛽+2𝛾𝑢1+3𝛿𝑢21), 𝑘 is a root of second order algebraic equation (2+𝛼𝜆)𝑖𝑘2(𝛾𝜆+3𝛿𝜆𝑢1+𝛼𝛼𝑢1)𝑘𝑖(𝛽+2𝛾𝑢1+3𝛿𝑢21)=0, 𝑢1=(𝛾±Δ)/2𝛿, 𝜉0, 𝐶0 are arbitrary constants.

By using of formulas (3.11) and (3.12), we can obtain multiple new periodic wave solutions in form tan𝜉 and cot𝜉.

Choosing the solutions (3.3) ((3.4), (3.13), (3.14), (3.18), (3.22), (3.23), (3.27) and (3.28), resp.) as a new “seed” solution 𝑢1(𝑥,𝑡) and solving the linear PDEs (2.11), (2.12), one gets a quasisolution 𝜙(𝑥,𝑡). Then substituting the quasisolution 𝜙(𝑥,𝑡) and 𝑢1(𝑥,𝑡) chosen above into (2.14), we can obtain more and more new exact particular solutions of (1.1). Taking 𝐶=1 in solutions (3.3), (3.4), (3.22), and (3.23), we can obtain shock wave solutions and singular traveling wave solutions of (1.1). Putting 𝐶=1 in solutions (3.9), (3.10), (3.27), and (3.28), we can obtain periodic wave solutions in form tan𝜉 and cot𝜉.

4. Conclusion

It is worthwhile pointing out that the exact solutions obtained in this paper have more general form than some known solutions in previous studies. In addition to rederiving all known solutions in a systematic way, several entirely new exact solutions can also be obtained. Specially, choosing 𝛼=0 in the all solutions above, one can obtain abundant explicit and exact solutions to the Kolmogorov-Petrovskii-Piskunov equation [16]. Setting 𝛼=0, 𝛾=0, 𝛽=𝛿 in the all solutions above, one can get abundant explicit and exact solutions to the Chaffee-Infante reaction diffusion equation [17]. We can also obtain abundant explicit and exact solutions to the Burgers-Huxley equation [18] by taking 𝛼0, 𝛽=𝛿𝜂, 𝛾=(1+𝜂)𝛿, 𝜂 arbitrary in the all solutions above. Go a step further, taking 𝛼=0, 𝛽=𝜂, 𝛾=(1+𝜂), 𝜂 arbitrary in the all solutions above, we also obtain abundant explicit and exact solutions to the FitzHugh-Nagumo equation [19]. We can obtain abundant explicit exact solutions to the Newell-Whitehead equation when taking 𝛼=0, 𝛽=1, 𝛾=0, 𝛿=1 in the all solutions above [17]. Putting 𝛼=0, 𝛽=1(3𝜂/2), 𝛾=(5𝜂/2)2, 𝛿=1𝜂 in the all solutions above, we can obtain abundant explicit and exact solutions to an isothermal autocatalytic system [20].

Acknowledgments

This work is supported by the National Science Foundation of China (10771041, 40890150, 40890153), the Scientific Program (2008B080701042) of Guangdong Province, China. The authors would like to thank Professor Wang Mingliang for his helpful suggestions.

References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar
  2. B. Grammaticos, A. Ramani, and J. Hietarinta, “A search for integrable bilinear equations: the Painlevé approach,” Journal of Mathematical Physics, vol. 31, no. 11, pp. 2572–2578, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. G. B. Whitham, “Comments on periodic waves and solitons,” IMA Journal of Applied Mathematics, vol. 32, no. 1–3, pp. 353–366, 1984. View at Publisher · View at Google Scholar
  4. G. B. Whitham, “On shocks and solitary waves,” Scripps Institution of Oceanography Reference Series, pp. 91–124, 1991.
  5. J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. D. Gibbon, A. C. Newell, M. Tabor, and Y. B. Zeng, “Lax pairs, Bäcklund transformations and special solutions for ordinary differential equations,” Nonlinearity, vol. 1, no. 3, pp. 481–490, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. Weiss, “Bäcklund transformation and the Painlevé property,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
  8. J. Hietarinta, “Hirota's bilinear method and partial integrability,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Boccara, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
  9. N. G. Berloff and L. N. Howard, “Solitary and periodic solutions of nonlinear nonintegrable equations,” Studies in Applied Mathematics, vol. 99, no. 1, pp. 1–24, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. L. Wang, “Exact solutions for a compound KdV-Burgers equation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. L. Wang, Y. B. Zhou, and H. Q. Zhang, “A nonlinear transformation of the shallow water wave equations and its application,” Advances in Mathematics, vol. 28, no. 1, pp. 72–75, 1999.
  13. E. G. Fan and H. Q. Zhang, “A new approach to Bäcklund transformations of nonlinear evolution equations,” Applied Mathematics and Mechanics, vol. 19, no. 7, pp. 645–650, 1998. View at Publisher · View at Google Scholar
  14. Y. D. Shang, “Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1286–1297, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. D. Shang, J. Qin, Y. Huang, and W. Yuan, “Abundant exact and explicit solitary wave and periodic wave solutions to the Sharma-Tasso-Olver equation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 532–538, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C. Roman and P. Oleksii, “New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations,” Journal of Physics A, vol. 40, no. 33, pp. 10049–10070, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. F. Zhang, “Exact and explicit solitary wave solutions to some nonlinear equations,” International Journal of Theoretical Physics, vol. 35, no. 8, pp. 1793–1798, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. A. M. Wazwaz, “Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 639–656, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. R. Cherniha, “New Q symmetries and exact solutions of some reaction-diffusion-convection equations arising in mathematical biology,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 783–799, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. J. H. Merkin, R. A. Satnoianu, and S. K. Scott, “Travelling waves in a differential flow reactor with simple autocatalytic kinetics,” Journal of Engineering Mathematics, vol. 33, no. 2, pp. 157–174, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH