Research Article | Open Access

Zitian Li, "Construction of New Exact Solutions for the (3 + 1)-Dimensional Burgers System", *Mathematical Problems in Engineering*, vol. 2014, Article ID 390742, 5 pages, 2014. https://doi.org/10.1155/2014/390742

# Construction of New Exact Solutions for the (3 + 1)-Dimensional Burgers System

**Academic Editor:**Miguel A. F. Sanjuan

#### Abstract

By means of a variable separation method and a generalized direct ansätz function approach, new exact solutions including cross kink-wave solutions, doubly periodic kinky-wave solutions, and breather type of two-solitary wave solutions for the (3 + 1)-dimensional Burgers system are obtained. Moreover, the mechanical features are also investigated.

#### 1. Introduction

Searching for explicit solutions of nonlinear evolution equations by using various different methods is useful and meaningful in physical science and nonlinear science. Many powerful methods have been presented, such as inverse scattering transform [1], Hirota’s bilinear form method [2], two-soliton method [3], homoclinic test technique [4], Bäcklund transformation method [5], and three-wave type of ansätz approach [6]. And much work has been focused on the various extensions and application of the known algebraic methods to construct the solutions of nonlinear evolution equations [7, 8].

In this letter, we will present exact solutions of the (3 + 1)-dimensional Burgers system [9]: which was derived from the inverse transformation of heat-conduction equation, where , , are three physical field function and . Recently, (1) was widely researched [10–12]. When , , (1) degenerated into the -dimensional Burgers equation and the soliton-like solutions were obtained by using the extended Riccati equation mapping method [13], and many other results were found in the literature [14, 15]. In our work, we will give some new exact solutions for (1) by using the variable separation hypothesis and the direct ansätz function method all together.

#### 2. Exact Solutions to the (3 + 1)-Dimensional Burgers System

We firstly suppose the solution of system (1) has the following ansätz: where , and are real constants and has the following variable separation form: with constants and are unknown function about and , respectively. Substituting (2) and (3) into (1), we have

Now let us consider (5), since is only a function about variables and , letting and by using the wave transformation , (5) can be changed into where , and are arbitrary constants. And when and are taken as , or , , we find that (4) is automatically satisfied.

When , solutions of (7) are well known as follows: where are arbitrary constants.

Therefore, we only need to solve (6) from which we can obtain the solutions of system (1).

*Example 1. *Let the test function be
where , , , , and are constants to be determined.

By introducing means of computer algebra (i.e., Maple), we obtain
which is a cross kink-wave solution, where , (see Figure 1: ).

**(a)**

**(b)**

*Example 2. *Similarly, if was taken as
where , and are constants to be determined later, then, we derived a periodic soliton solution and a multiple-soliton solution as follows:
where and (see Figure 1: , Figure 2: ).

**(a)**

**(b)**

*Example 3. *Furthermore, when choosing to be
with , and constants, we obtained a breather type of two-solitary wave solution as follows:
where , , .

The solution represented by (14) is breather type of two-solitary wave solution which contains a periodic wave and two solitary waves, whose amplitude periodically oscillates with the evolution of time (see Figure 2: ).

#### 3. Conclusion

In this paper, by using the variable separation method and the generalized direct ansätz method the -dimensional Burgers system is investigated. New exact solutions including cross kink-wave solution, doubly kinky-wave solution, and breather type of two-solitary wave solutions are obtained; these solutions enrich the structures of solutions of the -dimensional Burgers system. Moreover, the mechanical features are also investigated.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by Natural Science Foundation of Yunnan Province under Grant no. 2013FZ113.

#### References

- M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform*, Cambridge University Press, Cambridge, UK, 1991. View at: MathSciNet - M. Jimbo and T. Miwa, “Solitons and infinite dimensional Lie algebras,”
*Publications of the Research Institute for Mathematical Sciences*, vol. 19, no. 3, pp. 943–1001, 1983. View at: Publisher Site | Google Scholar | MathSciNet - Z. Dai, Z. Li, Z. Liu, and D. Li, “Exact cross kink-wave solutions and resonance for the Jimbo-Miwa equation,”
*Physica A*, vol. 384, no. 2, pp. 285–290, 2007. View at: Publisher Site | Google Scholar - Z. Dai, Z. Li, Z. Liu, and D. Li, “Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation,”
*Physics Letters A*, vol. 372, no. 17, pp. 3010–3014, 2008. View at: Publisher Site | Google Scholar | MathSciNet - R. Beals, M. Rabelo, and K. Tenenblat, “Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations,”
*Studies in Applied Mathematics*, vol. 81, no. 2, pp. 125–151, 1989. View at: Google Scholar | MathSciNet - Z. Li and Z. Dai, “Exact periodic cross-kink wave solutions and breather type of two-solitary wave solutions for the $(3+1)$-dimensional potential-{YTSF} equation,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 1939–1945, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Z. Li, “New exact homoclinic wave and periodic wave solutions for the Ginzburg-Landau equation,”
*Applied Mathematics and Computation*, vol. 217, no. 4, pp. 1549–1554, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Li and X. Zhang, “New exact kink solutions and periodic form solutions for a generalized Zakharov-Kuznetsov equation with variable coefficients,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 11, pp. 3418–3422, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-P. Ying and S.-Y. Lou, “Multilinear variable separation approach in (3+1)-dimensions: the Burgers equation,”
*Chinese Physics Letters*, vol. 20, no. 9, pp. 1448–1451, 2003. View at: Publisher Site | Google Scholar - L. Huang, J. A. Sun, F. Q. Dou, W. Duan, and X. X. Liu, “New variable separation solutions, localized structures and fractals of the $(3+1)$-dimensional nonlinear Burgers system,”
*Acta Physica Sinica*, vol. 56, no. 2, pp. 611–619, 2007. View at: Google Scholar | MathSciNet - S. H. Ma, X. H. Wu, J. P. Fang, and C. L. Zheng, “New exact solutions and special soliton structures of the $(3+1)$-dimensional Burgers system,”
*Acta Physica Sinica*, vol. 57, no. 1, pp. 11–17, 2008. View at: Google Scholar | MathSciNet - A.-M. Wazwaz, “Multiple soliton solutions and multiple singular soliton solutions for the (3 + 1)-dimensional Burgers equations,”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 942–948, 2008. View at: Publisher Site | Google Scholar | MathSciNet - F. Kong and S. Chen, “New exact soliton-like solutions and special soliton-like structures of the $(2+1)$ dimensional Burgers equation,”
*Chaos, Solitons and Fractals*, vol. 27, no. 2, pp. 495–500, 2006. View at: Publisher Site | Google Scholar | MathSciNet - A.-M. Wazwaz, “Multiple kink solutions and multiple singular kink solutions for the $(2+1)$-dimensional Burgers equations,”
*Applied Mathematics and Computation*, vol. 204, no. 2, pp. 817–823, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D.-S. Wang, H.-B. Li, and J. Wang, “The novel solutions of auxiliary equation and their application to the $(2+1)$-dimensional Burgers equations,”
*Chaos, Solitons & Fractals*, vol. 38, no. 2, pp. 374–382, 2008. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Zitian Li. 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.