On the exact solutions of a modified Kortweg de Vries type equation and higherorder modified Boussinesq equation with damping term
 Dimpho Millicent Mothibi^{1} and
 Chaudry Masood Khalique^{1}Email author
https://doi.org/10.1186/168718472013166
© Mothibi and Khalique; licensee Springer 2013
Received: 5 March 2013
Accepted: 24 May 2013
Published: 12 June 2013
Abstract
In this paper, we obtain exact solutions of two nonlinear evolution equations, namely the modified Kortweg de Vries equation and the higherorder modified Boussinesq equation with damping term. The method employed to obtain the exact solutions is the $({G}^{\prime}/G)$expansion method. Traveling wave solutions of three types are obtained and these are the solitary waves, periodic and rational.
Keywords
1 Introduction
It is well known that nonlinear evolution equations, such as (1) and (2), are widely used as models to describe physical phenomena in different fields of applied sciences such as plasma waves, solid state physics, plasma physics and fluid mechanics. One of the basic physical problems for these models is to obtain their exact solutions for the better understanding of nonlinear models [1–22]. In the last few decades, a variety of effective methods for finding exact solutions, such as the homogeneous balance method [3], the ansatz method [4, 5], the variable separation approach [6], the inverse scattering transform method [7], the Bäcklund transformation [8], the Darboux transformation [9], Hirota’s bilinear method [10], the reduction mKdV equation method [11], the trifunction method [12, 13], the projective Riccati equation method [14], the sinecosine method [15], the Jacobi elliptic function expansion method [16, 17], the Fexpansion method [18] and the expfunction expansion method [19] and many others, were successfully applied to nonlinear differential equations.
Although a great deal of research work has been devoted to finding different methods to solve nonlinear evolution equations, there is no unique method. In 2007 Wang et al. [20] proposed a new method referred to as the $({G}^{\prime}/G)$expansion method for finding traveling wave solutions of nonlinear evolution equations. This paper showed that the $({G}^{\prime}/G)$expansion method is an effective method for finding exact solutions of nonlinear evolution equations. It has been extensively used by various researchers (see, for example, papers [20–22]) in a variety of scientific fields. The key ideas of the method are that the traveling wave solutions of a complicated nonlinear evolution equation can be constructed by means of various solutions of a secondorder linear ordinary differential equation [20].
In this work, our main focus is on equations (1) and (2). We derive the traveling wave solutions of the two equations by using the $({G}^{\prime}/G)$expansion method. The paper is organized as follows. In Section 2, we describe the $({G}^{\prime}/G)$expansion method. Exact solutions of the modified Kortweg de Vries equation (1) and the higherorder modified Boussinesq equation with damping term (2) are constructed in Section 3 using the $({G}^{\prime}/G)$expansion method. In Section 4, conclusion is given.
2 Analysis of the $({G}^{\prime}/G)$expansion method
The $({G}^{\prime}/G)$expansion method for finding exact solutions of nonlinear differential equations was introduced in [20]. Several researchers have recently applied this method to various nonlinear differential equations. They have shown that this method provides a very effective and powerful mathematical tool for solving nonlinear equations in various fields of applied sciences (see, for example, papers [20–22]).
where $u(x,t)$ is an unknown function, P is a polynomial in u and its various partial derivatives, in which the highestorder derivatives and nonlinear terms are involved. The essence of the $({G}^{\prime}/G)$expansion method is given in the following steps.

Step 1. The transformation $u(x,t)=U(z)$, $z=x\nu t$ reduces equation (3) to the ordinary differential equation (ODE)$P(U,\nu {U}^{\prime},{U}^{\prime},{\nu}^{2}{U}^{\u2033},\nu {U}^{\u2033},{U}^{\u2033},\dots )=0.$(4)

Step 2. According to the $({G}^{\prime}/G)$expansion method, it is assumed that the traveling wave solution of equation (4) can be expressed by a polynomial in $({G}^{\prime}/G)$ as follows:$U(z)=\sum _{i=0}^{m}{\alpha}_{i}{\left(\frac{{G}^{\prime}}{G}\right)}^{i},$(5)
with ${\alpha}_{i}$, $i=0,1,2,\dots ,m$, λ and μ being constants to be determined. The positive integer m is determined by considering the homogenous balance between the highestorder derivatives and nonlinear terms appearing in ODE (4).

Step 3. By substituting (5) into (4) and using the secondorder ODE (6), collecting all terms with same order of $({G}^{\prime}/G)$ together, the lefthand side of (4) is converted into another polynomial in $({G}^{\prime}/G)$. Equating each coefficient of this polynomial to zero, yields a set of algebraic equations for ${\alpha}_{0},\dots ,{\alpha}_{m}$, ν, λ, μ.

Step 4. Assuming that the constants can be obtained by solving the algebraic equations in Step 3, since the general solution of (6) is known, then substituting the constants and the general solutions of (6) into (5) we obtain traveling wave solutions of the NPDE (3).
3 Exact solutions of (1) and (2)
In this section we construct traveling wave solutions of mKdV and modified Boussinesq equations by employing the $({G}^{\prime}/G)$expansion method.
3.1 The modified Kortweg de Vries equation
where u is a realvalued scalar function, t is time and x is a spatial variable.
where the prime denotes the derivative with respect to z.
where λ and μ are constants.
Substituting these values of ${\mathcal{A}}_{0}$, ${\mathcal{A}}_{1}$ and the corresponding solution of ODE (13) into (14), we obtain three types of traveling wave solutions of equation (7). These are as follows.
where $z=x\nu t$, ${\delta}_{1}=\frac{1}{2}\sqrt{{\lambda}^{2}4\mu}$, ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
where $z=x\nu t$, ${\delta}_{2}=\frac{1}{2}\sqrt{4\mu {\lambda}^{2}}$, ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
where $z=x\nu t$, ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
3.2 Higherorder modified Boussinesq equation with damping term
where u is a realvalued scalar function, t is time, x is a spatial variable and α, β, γ are nonzero real constants.
Substituting these values from (34) and the corresponding solution of ODE (13) into (27) yields three types of traveling wave solutions of equation (26) as follows.
where $z=x\nu t$, ${\delta}_{1}=\frac{1}{2}\sqrt{{\lambda}^{2}4\mu}$ and ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
where $z=x\nu t$, ${\delta}_{2}=\frac{1}{2}\sqrt{4\mu {\lambda}^{2}}$ and ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
where $z=x\nu t$, and ${C}_{1}$ and ${C}_{2}$ are arbitrary constants.
4 Conclusion
In this paper, we studied two nonlinear partial differential equations that appear in a variety of scientific fields. These are the modified Kortweg de Vries equation and the higherorder modified Boussinesq equation with damping term. We used the $({G}^{\prime}/G)$expansion method to obtain exact solutions of these two evolution equations. By using this method, we have successfully obtained traveling wave solutions expressed in the form of a hyperbolic function, a trigonometric function and a rational function. This work also highlighted the power of the $({G}^{\prime}/G)$expansion method for the determination of exact solutions of nonlinear evolution equations.
Declarations
Acknowledgements
DMM and CMK would like to thank the organizing Committee of the International Conference on the Theory, Methods and Application of Nonlinear Equations for their kind hospitality during the conference. DMM would also like to thank the Faculty Research Committee of FAST, NorthWest University for their financial support.
Authors’ Affiliations
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