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Yong Huang, Yadong Shang, "The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation", Abstract and Applied Analysis, vol. 2012, Article ID 489043, 11 pages, 2012. https://doi.org/10.1155/2012/489043
The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation
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.
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  and the Hirota bilinear method , 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 . 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 . 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  conjectured that these systems can be reduced to integrable equations. Another treatment of the partially integrable systems was offered by Hietarinta  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  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 . In [10–12], 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 . In a recent paper , 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 .
In this paper we investigate a general nonintegrable nonlinear convection-diffusion equation where , , , and 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
From (2.1), we have Substituting (2.1)–(2.5) into the left side of (1.1) and collecting all terms with , we obtain Setting the coefficient of in (2.6) to be zero, we obtain an ordinary differential equation for which has a solution where . And then By virtue of (2.7)–(2.9), (2.6) becomes Setting the coefficients of to be zero, respectively, it is easy to see from (2.10) that Substituting (2.8) into (2.1), we obtain a Bäcklund transformation where satisfy (2.11)–(2.13). Substituting a seed solution 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).
3. Exact Explicit Solutions to (1.1)
Noting the homogeneous property of (2.16) we can expect that in (2.16) is of the form with , , , , , and constants to be determined. Substituting (3.1) into (2.16), one gets a set of nonlinear algebraic equation Solving (3.2), we have the following.
Case 1. , , , and is a root of second-order algebraic equation .
Case 2. , , , and is a root of second-order algebraic equation .
Thus we obtain the following explicit exact solutions of (1.1) given by where , , is a root of second-order algebraic equation , , and are arbitrary constants.
We can also obtain the following explicit exact solutions of (1.1) given by where , , is a root of second-order algebraic equation , , and are arbitrary constants.
By direct computation, we readily obtain the following two useful formulas: where is arbitrary.
Analogously, we assume that in (2.16) is of the form with , , , , , and constants to be determined. Substituting (3.7) into (2.16), one gets a set of nonlinear algebraic equation Solving (3.8), we have the following.
Case 1. , , , and is a root of second order algebraic equation , .
Case 2. , , , and is a root of second order algebraic equation , .
Analogously, we have the following two useful formulas:
By virtue of the homogeneous property of (2.18), we can expect that is of the linear function form with and , constants to be determined. Substituting (3.15) into (2.18), we find that (3.15) satisfies (2.18), provided that and satisfy the following algebraic equations: where , , . Solving (3.16), we obtain that provided that coefficients , , and of (1.1) satisfy condition .
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: In order to obtain nontrivial solutions of (1.1), we need to require that , are all nonzero constants. Solving (3.19), one gets the following solutions.
Case 1. One has where is a root of second-order algebraic equation , , , .
Case 2. One has where is a root of second-order algebraic equation , , , .
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 equations In order to obtain a nontrivial solution of (1.1), we also need to assume that are all nonzero constants. Solving (3.24), we obtain the following.
Case 1. One has where is a root of second-order algebraic equation , , , .
Case 2. One has where is a root of second-order algebraic equation , , , .
Collecting (2.17), (3.7), (3.25), and (3.26), we obtain the following explicit exact periodic traveling wave solutions where , , , , is a root of second-order algebraic equation + + + , , , are arbitrary constants, where , , , , is a root of second order algebraic equation , , , are arbitrary constants.
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 and solving the linear PDEs (2.11), (2.12), one gets a quasisolution . Then substituting the quasisolution and chosen above into (2.14), we can obtain more and more new exact particular solutions of (1.1). Taking 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 in solutions (3.9), (3.10), (3.27), and (3.28), we can obtain periodic wave solutions in form and .
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 in the all solutions above, one can obtain abundant explicit and exact solutions to the Kolmogorov-Petrovskii-Piskunov equation . Setting , , in the all solutions above, one can get abundant explicit and exact solutions to the Chaffee-Infante reaction diffusion equation . We can also obtain abundant explicit and exact solutions to the Burgers-Huxley equation  by taking , , , arbitrary in the all solutions above. Go a step further, taking , , , arbitrary in the all solutions above, we also obtain abundant explicit and exact solutions to the FitzHugh-Nagumo equation . We can obtain abundant explicit exact solutions to the Newell-Whitehead equation when taking , , , in the all solutions above . Putting , , , in the all solutions above, we can obtain abundant explicit and exact solutions to an isothermal autocatalytic system .
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.
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