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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 [

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 this paper we investigate a general nonintegrable nonlinear convection-diffusion equation

According to the extended homogeneous balance method, we suppose that the solution of (

From (

Taking

Taking

In this section we want to obtain abundant exact explicit particular solutions of (

Noting the homogeneous property of (

Thus we obtain the following explicit exact solutions of (

We can also obtain the following explicit exact solutions of (

By direct computation, we readily obtain the following two useful formulas:

Thanks to the two formulas (

The solutions (

The solutions (

Analogously, we assume that

According to the result of Case

By the result of Case

Analogously, we have the following two useful formulas:

Due to the formula (

Owing to the formula (

By virtue of the homogeneous property of (

Substituting (

Now we suppose that (

One has

One has

By Case

According to the result of Case

According to formulas (

Analogously, we assume that (

One has

One has

Collecting (

By using of formulas (

Choosing the solutions (

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

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.