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# Dark soliton and periodic wave solutions of nonlinear evolution equations

- Özkan Güner
^{1}, - Ahmet Bekir
^{1}and - Adem Cengiz Cevikel
^{2}Email author

**2013**:68

https://doi.org/10.1186/1687-1847-2013-68

© Güner et al.; licensee Springer 2013

**Received:**8 October 2012**Accepted:**27 February 2013**Published:**21 March 2013

## Abstract

This paper studies the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. Subsequently, we formally derive the dark (topological) soliton solutions for these equations. By using the sine-cosine method, some additional periodic solutions are derived. The physical parameters in the soliton solutions of the *ansatz* method, amplitude, inverse width and velocity, are obtained as functions of the dependent model coefficients.

**PACS Codes:**02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fg.

## Keywords

- dark soliton
- sine-cosine method
*ansatz*method- Kudryashov-Sinelshchikov equation
- Jimbo-Miwa equation

## 1 Introduction

In recent years, many powerful methods to construct exact solutions of nonlinear partial differential equations have been established and developed, which led to one of the most exciting advances of nonlinear science and theoretical physics. Particularly, the existence of soliton-type solutions for nonlinear models is of great importance because of their potential application in many physics areas such as nonlinear optics, plasmas, fluid mechanics, condensed matter and many more. Remarkably, the interest in dark and bright solitons has been growing steadily in recent years [1–3]. In fact, many kinds of exact soliton solutions have been obtained by using, for example, the tanh-sech method [4–6], extended tanh method [7–9], homogeneous balance method [10, 11], first integral method [12, 13], Jacobi elliptic function method [14, 15], $(\frac{{G}^{\mathrm{\prime}}}{G})$-expansion method [16, 17], *F*-expansion method [18, 19], Hirota bilinear method [20, 21], multiple exp-function method [22] and transformed rational function method [23] and so on.

In this equation, *u* is a density and *α*, *b* are real parameters. Ryabov [26] obtained some exact solutions for $b=-3$ and $b=-4$ using a modification of the truncated expansion method. Solutions are derived in a more straightforward manner and cast into a simpler form, and some new types of solutions which contain solitary wave and periodic wave solutions are presented in [27].

This equation was first introduced by Jimbo and Miwa [28] and it is known that this model is not Painlevé integrable. For many years, many scientists have researched it and certain explicit solutions have been obtained [29–31]. Exact three-wave solutions including periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the Jimbo-Miwa equation have been obtained using the generalized three-wave method in [32].

The layout of this paper is organized as follows. In Section 2, we give the description of the sine-cosine method and we apply this method to the Kudryashov-Sinelshchikov (KS) and Jimbo-Miwa equations. We apply the *ansatz* method to the KS and JM equations in Section 3. Conclusions are given in the last section.

## 2 Sine-cosine method

In this section, the sine-cosine method will be first described and then subsequently applied to solve the Kudryashov-Sinelshchikov and Jimbo-Miwa equations.

### 2.1 Brief of the method

- 1.We introduce the wave variable $\xi =x+y-ct$ into the PDE$P(u,{u}_{t},{u}_{x},{u}_{y},{u}_{tt},{u}_{xx},{u}_{yy},{u}_{xt},{u}_{xy},{u}_{ty},\dots )=0,$(2.1)

- 2.The solutions of many nonlinear equations can be expressed in the form [34]$u(x,t)=\{\begin{array}{cc}\lambda {cos}^{\beta}(\mu \xi ),\hfill & |\xi |\le \frac{\pi}{2\mu},\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$(2.4)

*λ*,

*μ*and $\beta \ne 0$ are parameters that will be determined,

*μ*and

*c*are the wave number and the wave speed respectively. We use

and so on for other derivatives.

3. We substitute (2.6) or (2.7) into the reduced equation obtained above in (2.3), balance the terms of the cosine functions when (2.7) is used, or balance the terms of the sine functions when (2.6) is used, and solve the resulting system of algebraic equations by using the computerized symbolic calculations. We next collect all terms with the same power in ${cos}^{k}(\mu \xi )$ or ${sin}^{k}(\mu \xi )$ and set to zero their coefficients to get a system of algebraic equations among the unknowns *μ*, *β* and *λ*. We obtained all possible values of the parameters *μ*, *β* and *λ* [33].

### 2.2 Application of the sine-cosine method to the Kudryashov-Sinelshchikov equation

where *k* is the integration constant.

where $0<\sqrt{-\frac{\alpha}{8+4b}}(x-vt)<\pi $.

All the solutions reported in this paper have been verified with Maple by putting them back into original Eq. (1.1), which cannot be obtained by the methods [25–27]. To the best of our knowledge, these solutions are new and have not been reported yet.

### 2.3 Application of the sine-cosine method to the Jimbo-Miwa equation

where $0<\frac{1}{2}\sqrt{-2v-3}(x+y+z-vt)<\pi $.

Comparing the above results with the relevant ones in [28–31], it can be seen that some of the obtained results are new and the rest of solutions are the same.

## 3 *Ansatz* method

In this section, the *ansatz* method will be used to carry out the integration of the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. The search is going to be for a topological 1-soliton solution which is also known as a kink solution or a shock wave solution. This will be demonstrated in the following two subsections. For both equations, arbitrary constant coefficients will be considered. There are many applications of this method [38–44].

### 3.1 Application of the *ansatz* method to the Kudryashov-Sinelshchikov equation

*A*and

*B*are free parameters and

*v*is the velocity of the wave in (3.1) and (3.2). The exponent

*p*is unknown at this point and its values will fall out in the process of deriving the solution of this equation. Thus from (3.1) we get

*p*is yielded when the exponent pairs $2p-3$ and $p-1$, $2p+1$ and $p+3$ are equated with each other, respectively.

where the free parameter *A* is given by (3.18), the velocity of the solitons *v* is given in (3.19).

### 3.2 Application of the *ansatz* method to the Jimbo-Miwa equation

*A*and

*B*are free parameters and

*v*is the velocity of the wave in (3.21) and (3.22). The exponent

*p*is unknown at this point and its values will fall out in the process of deriving the solution of this equation. Thus from (3.21) we get

*p*is yielded when the exponent pairs $2p-3$ and $p-2$, $2p+1$ and $p+2$, $2p-1$ and

*p*are equated with each other, respectively.

where the free parameter *A* is given by (3.36) and the velocity of the solitons *v* is given in (3.36).

## 4 Conclusion

In this paper, the KS and JM equations are solved by the sine-cosine method as well as by the solitary wave *ansatz* method. There are several solutions that are obtained by the first method. The solitary wave *ansatz* method is used to carry out the integration of these equations. The obtained solutions may be useful for understanding of the mechanism of complicated nonlinear physical phenomena in wave interaction. In addition, we note that the solitary wave *ansatz* method is an efficient method for constructing exact soliton solutions for nonlinear wave equations. These results are going to be very useful for conducting research in future.

## Declarations

## Authors’ Affiliations

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## Copyright

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