## Advances in Nonlinear Complexity Analysis for Partial Differential Equations

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Zhenhui Xu, Hanlin Chen, Wei Chen, "The Multisoliton Solutions for the -Dimensional Sawada-Kotera Equation", *Abstract and Applied Analysis*, vol. 2013, Article ID 767254, 5 pages, 2013. https://doi.org/10.1155/2013/767254

# The Multisoliton Solutions for the -Dimensional Sawada-Kotera Equation

**Academic Editor:**Lan Xu

#### Abstract

Applying bilinear form and extended three-wavetype of ansätz approach on the -dimensional Sawada-Kotera equation, we obtain new multisoliton solutions, including the double periodic-type three-wave solutions, the breather two-soliton solutions, the double breather soliton solutions, and the three-solitary solutions. These results show that the high-dimensional nonlinear evolution equation has rich dynamical behavior.

#### 1. Introduction

As is well known that the exact solutions of nonlinear evolution equations play an important role in nonlinear science field, especially in nonlinear physical science since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications. The search for exact solutions of nonlinear partial differential equations has long been an interesting and hot topic in nonlinear mathematical physics. Consequently, many methods are available to look for exact solutions of nonlinear evolution equations, such as the inverse scattering method, the Lie group method, the mapping method, Exp-function method, and ansätz technique [1–4]. Very recently, Wang et al. [5] proposed a new technique called extended three-wave approach to seek multiwave solutions for integrable equations, and this method has been used to investigate several equations [6, 7]. In this paper, we consider the following Sawada-Kotera equation: Equation (1) was derived by B. G. Konopelchenko and V. G. Dubrovsky, and was called the Sawada-Kotera (SK) equation; for example, see [8]. By means of the two-soliton method, the exact periodic soliton solutions, N-soliton solutions, and traveling wave solutions of the SK equation were found [8–10].

In this paper, we discuss further the -dimensional SK equation, by using bilinear form and extended three-wave type of ansätz approach, respectively [5, 11–15], and some new multisoliton solutions are obtained.

#### 2. The Multisoliton Solutions

We assume where is an unknown real function. Substituting (2) into (1), we can reduce (1) into the following equation [8]: where the Hirota bilinear operator is defined by

Now we suppose the solution of (3) as where , , , and , , and () are some constants to be determined later. Substituting (5) into (3) and equating all the coefficients of different powers of , , , , , , and the constant term to zero, we can obtain a set of algebraic equations for , , , and (; ). Solving the system with the aid of Maple, we get the following results.

*Case 1. * If , then
where , , , and are free real constants. Substituting (6) into (5) and taking , we have
where , , , , , and . Substituting (7) into (2) yields the three-soliton solution of SK equation as follows:
where , , and .

If taking in (7), then we have
where , , , , , , and . Substituting (9) into (2) yields the double breather soliton solution of SK equation as follows:
where , , and .

*Case 2. *If , then
where , , , , , and are free real constants. Substituting (11) into (5) and taking , we have
Substituting (12) into (2) yields the breather two-soliton solution of SK equation as follows:
where , , and .

The expression is the breather two-soliton solution of SK equation which is a periodic wave in , and meanwhile is a two-soliton in , (refer to Figure 1(b)).

**(a)**

**(b)**

**(c)**

**(d)**

*Case 3. *If , then
where , , and are free real constants. Substituting (14) into (5) and taking , we have
where . Substituting (15) into (2) yields the breather two-soliton solution of SK equation as follows:
where , , , and .

The expression is the breather two-soliton solution of SK equation which is a periodic wave in and meanwhile is a two-soliton in , and in , respectively (refer to Figure 1(c)).

Notice that and are also the breather two-soliton solutions, but their structure is different, because the two wave propagation directions are different in the and , respectively (refer to Figures 1(b) and 1(c)).

If taking , in (12), then we have
when . Substituting (17) into (2) gives the double-periodic three-wave solution of SK equation as follows:
where , , and .

#### 3. Conclusion

By using bilinear form and extended three-wave type of ansätz approach, we discuss further the -dimensional Sawada-Kotera equation and find some new multisoliton solutions. The result shows that the extended three-wave type of ansätz approach may provide us with a straightforward and effective mathematical tool for seeking multiwave solutions of high-dimensional nonlinear evolution equations.

#### Acknowledgments

This work was supported by Chinese Natural Science Foundation Grant nos. 11061028, 10971169. Sichuan Educational Science Foundation Grant no. 09zc008.

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#### Copyright

Copyright © 2013 Zhenhui Xu et al. 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.