Dark soliton and periodic wave solutions of nonlinear evolution equations

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.

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 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]:

$$u_t+\alpha u{u}_x+u_{xxx}-{(u{u}_{xx})}_x-b{u}_x{u}_{xx}=0.$$

(1.1)

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 Jimbo-Miwa equation (JM)

$$u_{xxxy}+3u_y{u}_{xx}+3u_x{u}_{xy}+2u_{yt}-3u_{xz}=0.$$

(1.2)

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

The sine-cosine 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 $\xi =x+y-ct$ 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)

where $u(x,y,t)$ is a traveling wave solution. This enables us to use the following changes:

$$\begin{array}{r}\frac{\partial }{\partial t}=-c\frac{\partial }{\partial \xi },\frac{{\partial }^2}{\partial t^2}=c^2\frac{{\partial }^2}{\partial {\xi }^2},\frac{\partial }{\partial x}=\frac{\partial }{\partial \xi },\\ \frac{{\partial }^2}{\partial x^2}=\frac{{\partial }^2}{\partial {\xi }^2},\frac{\partial }{\partial y}=\frac{\partial }{\partial \xi },\frac{{\partial }^2}{\partial y^2}=\frac{{\partial }^2}{\partial {\xi }^2},\dots .\end{array}$$

(2.2)

One can immediately reduce nonlinear PDE (2.1) into the nonlinear ODE

$$Q(u,u_{\xi },u_{\xi \xi },u_{\xi \xi \xi },\dots )=0.$$

(2.3)

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

$$u(x,t)=\{\begin{array}{cc}\lambda {sin}^{\beta }(\mu \xi ),\hfill & |\xi |\le \frac{\pi }{\mu },\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$$

(2.5)

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

$$\begin{array}{r}u(\xi )=\lambda {cos}^{\beta }(\mu \xi ),\\ u^n(\xi )={\lambda }^n{cos}^{n\beta }(\mu \xi ),\\ {\left(u^n\right)}_{\xi }=-n\mu \beta {\lambda }^{n}sin(\mu \xi ){cos}^{n\beta -1}(\mu \xi ),\\ {\left(u^n\right)}_{\xi \xi }=-n^2{\mu }^2{\beta }^2{\lambda }^n{cos}^{n\beta }(\mu \xi )+n{\mu }^2{\lambda }^n\beta (n\beta -1){cos}^{n\beta -2}(\mu \xi ),\end{array}$$

(2.6)

and the derivatives of (2.5) become

$$\begin{array}{r}u(\xi )=\lambda {sin}^{\beta }(\mu \xi ),\\ u^n(\xi )={\lambda }^n{sin}^{n\beta }(\mu \xi ),\\ {\left(u^n\right)}_{\xi }=n\mu \beta {\lambda }^{n}cos(\mu \xi ){sin}^{n\beta -1}(\mu \xi ),\\ {\left(u^n\right)}_{\xi \xi }=-n^2{\mu }^2{\beta }^2{\lambda }^n{sin}^{n\beta }(\mu \xi )+n{\mu }^2{\lambda }^n\beta (n\beta -1){sin}^{n\beta -2}(\mu \xi ),\end{array}$$

(2.7)

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

We begin first with Eq. (1.1). Using the wave variable $\xi =x-vt$, Eq. (1.1) is carried to the ODE

$$-{(uv)}^{\mathrm{\prime }}+\alpha {\left(\frac{u^2}{2}\right)}^{\mathrm{\prime }}+u^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}-{\left(u{u}^{\mathrm{\prime }\mathrm{\prime }}\right)}^{\mathrm{\prime }}-\frac{b}{2}{\left[{\left(u^{\mathrm{\prime }}\right)}^2\right]}^{\mathrm{\prime }}=0,$$

(2.8)

where by integrating once we obtain

$$-uv+\alpha \frac{u^2}{2}+u^{\mathrm{\prime }\mathrm{\prime }}-u{u}^{\mathrm{\prime }\mathrm{\prime }}-\frac{b}{2}{\left(u^{\mathrm{\prime }}\right)}^2+k=0,$$

(2.9)

where k is the integration constant.

Substituting (2.6) into (2.9) gives

(2.10)

Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:

$$\begin{array}{r}(\beta -1)\ne 0,\\ \beta -2=2\beta ,\\ \frac{1}{2}\alpha {\lambda }^2+4{\lambda }^2{\mu }^2+2b{\lambda }^2{\mu }^2=0,\\ -v\lambda -4\lambda {\mu }^2-2{\lambda }^2{\mu }^2-2b{\lambda }^2{\mu }^2=0,\\ 2\lambda {\mu }^2+k=0.\end{array}$$

(2.11)

Solving the system (2.11) yields

$$\begin{array}{r}\beta =-2,\\ \mu =\sqrt{-\frac{\alpha }{8+4b}},\\ \lambda =\frac{2k(b+2)}{\alpha },\\ v=\frac{a+2k+3kb+b^{2}k}{b+2}.\end{array}$$

(2.12)

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:

$$u(x,t)=\left(\frac{2k(b+2)}{\alpha }\right){sec}^2[\sqrt{-\frac{\alpha }{8+4b}}(x-vt)],$$

(2.13)

where $|\sqrt{-\frac{\alpha }{8+4b}}(x-vt)|<\frac{\pi }{2}$, and

$$u(x,t)=\left(\frac{2k(b+2)}{\alpha }\right){csc}^2[\sqrt{-\frac{\alpha }{8+4b}}(x-vt)],$$

(2.14)

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

However, for $\frac{\alpha }{8+4b}>0$, we obtain the soliton solutions

$$u(x,t)=\left(\frac{2k(b+2)}{\alpha }\right){sech}^2[\sqrt{-\frac{\alpha }{8+4b}}(x-vt)]$$

(2.15)

and

$$u(x,t)=(-\frac{2k(b+2)}{\alpha }){csch}^2[\sqrt{\frac{\alpha }{8+4b}}(x-vt)].$$

(2.16)

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

We begin second with Eq. (1.2). Using the wave variable $\xi =x+y+z-vt$, Eq. (1.2) is carried to the ODE

$$u^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+3{\left[{\left(u^{\mathrm{\prime }}\right)}^2\right]}^{\mathrm{\prime }}-(2v+3)u^{\mathrm{\prime }\mathrm{\prime }}=0,$$

(2.17)

where by integrating once we obtain

$$u^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}+3{\left(u^{\mathrm{\prime }}\right)}^2-(2v+3)u^{\mathrm{\prime }}=0,$$

(2.18)

which is obtained upon setting the constant of integration to zero. Setting $u^{\mathrm{\prime }}=\rho$, Eq. (2.18) becomes

$${\rho }^{\mathrm{\prime }\mathrm{\prime }}+3{\rho }^2-(2v+3)\rho =0.$$

(2.19)

Substituting (2.4) into (2.19) gives

(2.20)

Equating the exponents and the coefficients of each pair of the cosine functions, we find the following system of algebraic equations:

$$\begin{array}{r}(\beta -1)\ne 0,\\ \beta -2=2\beta ,\\ 6\lambda {\mu }^2+3{\lambda }^2=0,\\ -4\lambda {\mu }^2-2\lambda v-3\lambda =0.\end{array}$$

(2.21)

Solving the system (2.21) yields

$$\begin{array}{r}\beta =-2,\\ \mu =\frac{1}{2}\sqrt{-2v-3},\\ \lambda =v+\frac{3}{2}.\end{array}$$

(2.22)

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:

$$u(x,y,z,t)=(v+\frac{3}{2}){sec}^2[\frac{1}{2}\sqrt{-2v-3}(x+y+z-vt)],$$

(2.23)

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

$$u(x,y,z,t)=(v+\frac{3}{2}){csc}^2[\frac{1}{2}\sqrt{-2v-3}(x+y+z-vt)],$$

(2.24)

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

However, for $2v+3>0$, we obtain the soliton solutions

$$u(x,y,z,t)=(v+\frac{3}{2}){sech}^2[\frac{1}{2}\sqrt{-2v-3}(x+y+z-vt)]$$

(2.25)

and

$$u(x,y,z,t)=(-v-\frac{3}{2}){csch}^2[\frac{1}{2}\sqrt{2v+3}(x+y+z-vt)].$$

(2.26)

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

In this section the search is going to be for a topological 1-soliton solution to the Kudryashov-Sinelshchikov equation given by (1.1). To start off, the hypothesis is given by [45, 46]

(3.1)

(3.2)

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

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

Substituting Eqs. (3.3)-(3.7) into (1.1), we have

(3.8)

From (3.8), equating the exponents $2p-1$ and $p+1$ gives

$$2p-1=p+1$$

so that

$$p=2.$$

(3.9)

It should be noted that the same value of p is yielded when the exponent pairs $2p-3$ and $p-1$, $2p+1$ and $p+3$ are equated with each other, respectively.

(3.10)

(3.11)

(3.12)

(3.13)

If we put $p=2$ in (3.10)-(3.13), the system reduces to

(3.14)

(3.15)

(3.16)

Solving the above equations yields

(3.17)

(3.18)

(3.19)

Hence, finally, the 1-soliton solution to (1.1) is respectively given by

$$u(x,t)=A{tanh}^2[B(x-vt)],$$

(3.20)

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

In this section the search is going to be for a topological 1-soliton solution to the $(3+1)$-dimensional Jimbo-Miwa equation. Without any loss of generality, it is assumed that the dark soliton solution to (1.2) is given by

$$u(x,y,z,t)=A{tanh}^p\tau ,$$

(3.21)

where

$$\tau =B(x+y+z-vt).$$

(3.22)

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

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

Substituting Eqs. (3.23)-(3.31) into (1.2), we have

(3.32)

From (3.32), equating the exponents $2p+3$ and $p+4$ gives

$$2p+3=p+4$$

so that

$$p=1.$$

It should be noted that the same value of 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.

(3.33)

(3.34)

(3.35)

Solving the above system for $p=1$ gives

$$\begin{array}{r}A=2B,\\ v=2B^2-\frac{3}{2}.\end{array}$$

(3.36)

Thus, finally, the 1-soliton solution to (1.2) is respectively given by

$$u(x,y,z,t)=Atanh[B(x+y+z-vt)],$$

(3.37)

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.