 Research
 Open Access
 Published:
Dark soliton and periodic wave solutions of nonlinear evolution equations
Advances in Difference Equations volume 2013, Article number: 68 (2013)
Abstract
This paper studies the KudryashovSinelshchikov and JimboMiwa equations. Subsequently, we formally derive the dark (topological) soliton solutions for these equations. By using the sinecosine 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.
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 solitontype 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 tanhsech 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], Fexpansion method [18, 19], Hirota bilinear method [20, 21], multiple expfunction method [22] and transformed rational function method [23] and so on.
In 2010, Kudryashov and Sinelshchikov [24] obtained a more common nonlinear partial differential equation for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of the liquid and the heat transfer. The equation reads as follows [25]:
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].
On the other hand, there is the JimboMiwa equation (JM)
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 threewave solutions including periodic crosskink wave solutions, doubly periodic solitary wave solutions and breather type of twosolitary wave solutions for the JimboMiwa equation have been obtained using the generalized threewave method in [32].
The layout of this paper is organized as follows. In Section 2, we give the description of the sinecosine method and we apply this method to the KudryashovSinelshchikov (KS) and JimboMiwa equations. We apply the ansatz method to the KS and JM equations in Section 3. Conclusions are given in the last section.
2 Sinecosine method
In this section, the sinecosine method will be first described and then subsequently applied to solve the KudryashovSinelshchikov and JimboMiwa equations.
2.1 Brief of the method
The sinecosine method was first proposed by Wazwaz in 2004 [33]. This method has been applied to various kinds of nonlinear problems arising in the applied sciences [34–37].

1.
We introduce the wave variable $\xi =x+yct$ 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)
where $u(x,y,t)$ is a traveling wave solution. This enables us to use the following changes:
One can immediately reduce nonlinear PDE (2.1) into the nonlinear ODE
Ordinary differential equation (2.3) is then integrated as long as all terms contain derivatives, where we neglect integration constants.

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)
or in the form
where 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 the derivatives of (2.5) become
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 sinecosine method to the KudryashovSinelshchikov equation
We begin first with Eq. (1.1). Using the wave variable $\xi =xvt$, Eq. (1.1) is carried to the ODE
where by integrating once we obtain
where k is the integration constant.
Substituting (2.6) into (2.9) gives
Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:
Solving the system (2.11) yields
The result (2.11) can be easily obtained if we also use the sine method (2.5). Consequently, for $\frac{\alpha}{8+4b}<0$, the following periodic solutions can be obtained:
where $\sqrt{\frac{\alpha}{8+4b}}(xvt)<\frac{\pi}{2}$, and
where $0<\sqrt{\frac{\alpha}{8+4b}}(xvt)<\pi $.
However, for $\frac{\alpha}{8+4b}>0$, we obtain the soliton solutions
and
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 sinecosine method to the JimboMiwa equation
We begin second with Eq. (1.2). Using the wave variable $\xi =x+y+zvt$, Eq. (1.2) is carried to the ODE
where by integrating once we obtain
which is obtained upon setting the constant of integration to zero. Setting ${u}^{\mathrm{\prime}}=\rho $, Eq. (2.18) becomes
Substituting (2.4) into (2.19) gives
Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:
Solving the system (2.21) yields
The result (2.21) can be easily obtained if we also use the sine method (2.5). Consequently, for $2v+3<0$, the following periodic solutions can be obtained:
where $\frac{1}{2}\sqrt{2v3}(x+y+zvt)<\frac{\pi}{2}$, and
where $0<\frac{1}{2}\sqrt{2v3}(x+y+zvt)<\pi $.
However, for $2v+3>0$, we obtain the soliton solutions
and
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 KudryashovSinelshchikov and JimboMiwa equations. The search is going to be for a topological 1soliton 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 KudryashovSinelshchikov equation
In this section the search is going to be for a topological 1soliton solution to the KudryashovSinelshchikov equation given by (1.1). To start off, the hypothesis is given by [45, 46]
Here, 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
Substituting Eqs. (3.3)(3.7) into (1.1), we have
From (3.8), equating the exponents $2p1$ and $p+1$ gives
so that
It should be noted that the same value of p is yielded when the exponent pairs $2p3$ and $p1$, $2p+1$ and $p+3$ are equated with each other, respectively.
If we put $p=2$ in (3.10)(3.13), the system reduces to
Solving the above equations yields
Hence, finally, the 1soliton solution to (1.1) is respectively given by
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 JimboMiwa equation
In this section the search is going to be for a topological 1soliton solution to the $(3+1)$dimensional JimboMiwa equation. Without any loss of generality, it is assumed that the dark soliton solution to (1.2) is given by
where
Here, 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
Substituting Eqs. (3.23)(3.31) into (1.2), we have
From (3.32), equating the exponents $2p+3$ and $p+4$ gives
so that
It should be noted that the same value of p is yielded when the exponent pairs $2p3$ and $p2$, $2p+1$ and $p+2$, $2p1$ and p are equated with each other, respectively.
Solving the above system for $p=1$ gives
Thus, finally, the 1soliton solution to (1.2) is respectively given by
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 sinecosine 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.
References
 1.
Triki H, Wazwaz AM: Bright and dark soliton solutions for a $K(m,n)$equation with t dependent coefficients. Phys. Lett. A 2009, 373: 21622165. 10.1016/j.physleta.2009.04.029
 2.
Green PD, Biswas A: Bright and dark optical solitons with timedependent coefficients in a nonKerr law media. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 38653873. 10.1016/j.cnsns.2010.01.018
 3.
Biswas A, Milovic D: Bright and dark solitons of the generalized nonlinear Schrödinger’s equation. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 14731484. 10.1016/j.cnsns.2009.06.017
 4.
Malfliet W: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 1992, 60: 650654. 10.1119/1.17120
 5.
Malfliet W, Hereman W: The tanh method. I: exact solutions of nonlinear evolution and wave equations. Phys. Scr. 1996, 54: 563568. 10.1088/00318949/54/6/003
 6.
Wazwaz AM: The tanh method for travelling wave solutions of nonlinear equations. Appl. Math. Comput. 2004, 154(3):713723. 10.1016/S00963003(03)007458
 7.
Abdou MA: New solitons and periodic wave solutions for nonlinear physical models. Nonlinear Dyn. 2008, 52(12):129136. 10.1007/s1107100792657
 8.
Fan E: Extended tanhfunction method and its applications to nonlinear equations. Phys. Lett. A 2000, 277: 212218. 10.1016/S03759601(00)007258
 9.
Wazwaz AM: The extended tanh method for new soliton solutions for many forms of the fifthorder KdV equations. Appl. Math. Comput. 2007, 184(2):10021014. 10.1016/j.amc.2006.07.002
 10.
Fan E, Zhang H: A note on the homogeneous balance method. Phys. Lett. A 1998, 246: 403406. 10.1016/S03759601(98)005477
 11.
Wang ML: Exact solutions for a compound KdVBurgers equation. Phys. Lett. A 1996, 213: 279287. 10.1016/03759601(96)00103X
 12.
Feng ZS: The first integral method to study the BurgersKdV equation. J. Phys. A, Math. Gen. 2002, 35: 343. 10.1088/03054470/35/2/312
 13.
Taghizadeh N, Mirzazadeh M: The first integral method to some complex nonlinear partial differential equations. J. Comput. Appl. Math. 2011, 235(16):4871. 10.1016/j.cam.2011.02.021
 14.
Fan E, Zhang J: Applications of the Jacobi elliptic function method to specialtype nonlinear equations. Phys. Lett. A 2002, 305: 383392. 10.1016/S03759601(02)015165
 15.
Liu S, Fu Z, Liu S, Zhao Q: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289: 6974. 10.1016/S03759601(01)005801
 16.
Wang ML, Li X, Zhang J:The $(\frac{{G}^{\mathrm{\prime}}}{G})$expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372(4):417423. 10.1016/j.physleta.2007.07.051
 17.
Bekir A:Application of the $(\frac{{G}^{\mathrm{\prime}}}{G})$expansion method for nonlinear evolution equations. Phys. Lett. A 2008, 372(19):34003406. 10.1016/j.physleta.2008.01.057
 18.
Abdou MA: Further improved F expansion and new exact solutions for nonlinear evolution equations. Nonlinear Dyn. 2008, 52(3):277288. 10.1007/s1107100792773
 19.
Zhang JL, Wang ML, Wang YM, Fang ZD: The improved F expansion method and its applications. Phys. Lett. A 2006, 350: 103109. 10.1016/j.physleta.2005.10.099
 20.
Wazwaz AM: Multiple soliton solutions for coupled KdV and coupled KP systems. Can. J. Phys. 2010, 87(12):12271232.
 21.
Wazwaz AM: Extended KP equations and extended system of KP equations: multiplesoliton solutions. Can. J. Phys. 2011, 89(7):739743. 10.1139/p11065
 22.
Ma WX, Huang T, Zhang Y: A multiple expfunction method for nonlinear differential equations and its application. Phys. Scr. 2010., 82: Article ID 065003
 23.
Ma WX:Comment on the $(3+1)$dimensional KadomtsevPetviashvili equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 2663. 10.1016/j.cnsns.2010.10.003
 24.
Kudryashov NA, Sinelshchikov DI: Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer. Phys. Lett. A 2010, 374: 20112016. 10.1016/j.physleta.2010.02.067
 25.
He B, Meng Q, Zhang J, Long Y: Periodic loop solutions and their limit forms for the KudryashovSinelshchikov equation. Math. Probl. Eng. 2012., 2012: Article ID 320163
 26.
Ryabov PN: Exact solutions of the Kudryashov and Sinelshchikov equation. Appl. Math. Comput. 2010, 217: 35853590. 10.1016/j.amc.2010.09.003
 27.
Randrüüt M: On the KudryashovSinelshchikov equation for waves in bubbly liquids. Phys. Lett. A 2011, 375: 36873692. 10.1016/j.physleta.2011.08.048
 28.
Jimbo M, Miwa T: Solitons and infinite dimensional Lie algebras. Publ. Res. Inst. Math. Sci. 1983, 19: 9431001. 10.2977/prims/1195182017
 29.
Senthilvelan M: On the extended applications of homogenous balance method. Appl. Math. Comput. 2001, 123: 381388. 10.1016/S00963003(00)00076X
 30.
Tang XY, Liang ZF:Variable separation solutions for the $(3+1)$dimensional JimboMiwa equation. Phys. Lett. A 2006, 351: 398402. 10.1016/j.physleta.2005.11.035
 31.
Wazwaz AM: New solutions of distinct physical structures to highdimensional nonlinear evolution equations. Appl. Math. Comput. 2008, 196: 363370. 10.1016/j.amc.2007.06.002
 32.
Li Z, Dai Z, Liu J:Exact threewave solutions for the $(3+1)$dimensional JimboMiwa equation. Comput. Math. Appl. 2011, 61: 20622066. 10.1016/j.camwa.2010.08.070
 33.
Wazwaz AM: A sinecosine method for handling nonlinear wave equations. Math. Comput. Model. 2004, 40: 499508. 10.1016/j.mcm.2003.12.010
 34.
Wazwaz AM: The tanh method and the sinecosine method for solving the KPMEW equation. Int. J. Comput. Math. 2005, 82(2):235246. 10.1080/00207160412331296706
 35.
Bekir A, Cevikel AC: New solitons and periodic solutions for nonlinear physical models in mathematical physics. Nonlinear Anal., Real World Appl. 2010, 11(4):32753285. 10.1016/j.nonrwa.2009.10.015
 36.
Bekir A: New solitons and periodic wave solutions for some nonlinear physical models by using sinecosine method. Phys. Scr. 2008, 77(4):501505.
 37.
Zedan HA, Monaquel SJ: The sinecosine method for the DaveyStewartson equations. Appl. Math. ENotes 2010, 10: 103111.
 38.
Triki H, Wazwaz AM:A onesoliton solution of the $ZK(m,n,k)$ equation with generalized evolution and timedependent coefficients. Nonlinear Anal., Real World Appl. 2011, 12: 28222825. 10.1016/j.nonrwa.2011.04.008
 39.
Triki H, Wazwaz AM: Bright soliton solution to a generalized BurgersKdV equation with timedependent coefficients. Appl. Math. Comput. 2010, 217: 466471. 10.1016/j.amc.2010.05.078
 40.
Triki H, Wazwaz AM: Bright and dark solitons for a generalized Korteweg deVriesmodified Korteweg deVries equation with higherorder nonlinear terms and timedependent coefficients. Can. J. Phys. 2011, 89(3):253259. 10.1139/P11015
 41.
Triki H, Wazwaz AM: Soliton solutions for a generalized KdV and BBM equations with timedependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 11221126. 10.1016/j.cnsns.2010.06.024
 42.
Biswas A:1soliton solution of the $B(m,n)$ equation with generalized evolution. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 32263229. 10.1016/j.cnsns.2008.12.025
 43.
Triki H, Ismail MS:Soliton solutions of a $BBM(m,n)$ equation with generalized evolution. Appl. Math. Comput. 2010, 217(1):4854. 10.1016/j.amc.2010.05.063
 44.
Esfahani A: On the generalized KadomtsevPetviashvili equation with generalized evolution and variable coefficients. Phys. Lett. A 2010, 374: 36353645. 10.1016/j.physleta.2010.07.015
 45.
Biswas A, Triki H, Labidi M: Bright and dark solitons of the RosenauKawahara equation with power law nonlinearity. Phys. Wave Phenom. 2011, 19(1):2429. 10.3103/S1541308X11010067
 46.
Triki H, Wazwaz AM: Dark solitons for a combined potential KdV and Schwarzian KdV equations with t dependent coefficients and forcing term. Appl. Math. Comput. 2011, 217: 88468851. 10.1016/j.amc.2011.03.050
Author information
Additional information
Competing interests
The authors declare that there is no conflict of interest among the authors.
Authors’ contributions
AÇ carried out the theoretical studies. AB carried out the computational studies and wrote the paper according to the rules of journal. ÖG found out the problem in literature, and carried out the computational studies. All authors read and approved the final manuscript.
Rights and permissions
About this article
Received
Accepted
Published
DOI
Keywords
 dark soliton
 sinecosine method
 ansatz method
 KudryashovSinelshchikov equation
 JimboMiwa equation