- Open Access
The modified alternative (G’/G)-expansion method to nonlinear evolution equation: application to the (1+1)-dimensional Drinfel’d-Sokolov-Wilson equation
© Akbar et al.; licensee Springer. 2013
- Received: 1 March 2013
- Accepted: 20 June 2013
- Published: 18 July 2013
Over the years, (G’/G)–expansion method is employed to generate traveling wave solutions to various wave equations in mathematical physics. In the present paper, the alternative (G’/G)–expansion method has been further modified by introducing the generalized Riccati equation to construct new exact solutions. In order to illustrate the novelty and advantages of this approach, the (1+1)-dimensional Drinfel’d-Sokolov-Wilson (DSW) equation is considered and abundant new exact traveling wave solutions are obtained in a uniform way. These solutions may be imperative and significant for the explanation of some practical physical phenomena. It is shown that the modified alternative (G’/G)–expansion method an efficient and advance mathematical tool for solving nonlinear partial differential equations in mathematical physics.
- (G’/G)-expansion method
- Travelling wave solutions
- DSW equation
- Nonlinear evolution equations
After the observation of soliton phenomena by John Scott Russell in 1834 (Wazwaz2009) and since the KdV equation was solved by Gardner et al. (1967) by inverse scattering method, finding exact solutions of nonlinear evolution equations (NLEEs) has turned out to be one of the most exciting and particularly active areas of research. The appearance of solitary wave solutions in nature is quite common. Bell-shaped sech-solutions and kink-shaped tanh-solutions model wave phenomena in elastic media, plasmas, solid state physics, condensed matter physics, electrical circuits, optical fibers, chemical kinematics, fluids, bio-genetics etc. The traveling wave solutions of the KdV equation and the Boussinesq equation which describe water waves are well-known examples. Apart from their physical relevance, the closed-form solutions of NLEEs if available facilitate the numerical solvers in comparison, and aids in the stability analysis. In soliton theory, there are several techniques to deal with the problems of solitary wave solutions for NLEEs, such as, Hirota’s bilinear transformation (Hirota1971), Backlund transformation (Rogers & Shadwick1982), improved homotopy perturbation (Jafari & Aminataei2010), Darboux transformation (Zhaqilao2010), tanh-function (Malfliet1992), homogeneous balance (Wang1996), Jacobi elliptic function (Liu et al.2001; Ali2011), F-expansion (Zhou et al.2003) and Exp-function (He & Wu2006; Abdou et al.2007; Akbar & Ali2011; Naher et al.2012). It is to be highlighted that Marinca and Herişanu (2011) applied a new approach for calculating a kind of explicit exact solution of nonlinear differential equations and in the simiar context obtained exact solutions of the Duffing and double-well Duffing equations. They implemented the new proposed procedure by using a quotient trigonometric function expansion method and also proved that the introduced method could be easily applied to solve other nonlinear differential equations.
Recently, Wang et al. (2008) established a widely used direct and concise method called the (G’/G)-expansion method for obtaining the exact travelling wave solutions of NLEEs, where G(ξ) satisfies the second order linear ordinary differential equation (ODE) G″ + λ G′ + μG = 0, where λ and μ are arbitrary constants. Applications of the (G’/G)-expansion method can be found in the articles (Bekir2008; Naher et al.2011; Akbar et al.2012; Kol & Tabi2011; Zayed & Gepreel2009; Zayed2009a; Zhang et al.2008a; Zhang et al.2008b; Abazari2010; Liu et al.2010) for better understanding.
In order to establish the effectiveness and reliability of the (G’/G)-expansion method and to expand the possibility of its application, further research has been carried out by several researchers. For instance, Zhang et al. (2010) presented an improved (G’/G)-expansion method to seek more general traveling wave solutions. Zayed (2009b) presented a new approach of the (G’/G)-expansion method where G(ξ) satisfies the Jacobi elliptic equation [G′(ξ)]2 = e 2 G 4(ξ) + e 1 G 2(ξ) + e 0, e 2, e 1, e 0 are arbitrary constants, and obtained new exact solutions. Zayed (2011) again presented an alternative approach of this method in which G(ξ) satisfies the Riccati equation G′(ξ) = A + B G 2(ξ), where A and B are arbitrary constants.
Still, substantial work has to be done in order for the (G’/G)-expansion method to be well established, since every nonlinear equation has its own physically significant rich structure. For finding the new exact solutions of NLEEs, it is important to present various method and ansatz, but it seems to be more important how to obtain more new exact solutions to NLEEs under the known method and ansatz. In the present article, we further modify the alternative (G’/G)-expansion method (presented by Zayed (2011)) by introducing the generalized Riccati equation mapping, its twenty seven solutions and constructed abundant new traveling wave solutions of the DSW equation.
where u=u(x, t) is an unknown function, P is a polynomial in u(x, t) and its partial derivatives in which the highest order partial derivatives and the nonlinear terms are involved. The main steps of the modified alternative (G’/G)-expansion method combined with the generalized Riccati equation mapping are as follows:
where the superscripts stands for the ordinary derivatives with respect to ξ.
where a n (n = 0, 1, 2, ⋯, m), r, p and q are arbitrary constants to be determined later.
where d is an arbitrary constant.
where c 1 is an arbitrary constant.
Step 3: To determine the positive integer m, substitute Equation (4) along with Equation (5) into Equation (3) and then consider homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Equation (3).
Step 4: Substituting Equation (4) along with Equation (5) into Equation (3) together with the value of m obtained in step 3, we obtain polynomials in G i and G –i (i = 0, 1, 2, 3 ⋯) and vanishing each coefficient of the resulted polynomial to zero, yields a set of algebraic equations for a n p, q, r and V.
Step 5: Suppose the value of the constants a n p, q, r and V can be determined by solving the set of algebraic equations obtained in step 4. Since the general solutions of Equation (5) are known, substituting, a n p, q, r and V into Equation (4), we obtain new exact traveling wave solutions of the nonlinear evolution Equation (1).
where a i , (i = 0, 1, 2, ⋯, m) and b j , (j = 0, 1, 2, ⋯, n) all are constants to be determined and G’/G(ξ) satisfies the generalized Riccati Equation (5). Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Equations (8) and (9), we obtain m=2 and n=1.
where b 1, p, q and r are arbitrary constants.
where c 1 is an arbitrary constant.
Because of the arbitrariness of the parameters b 1, p, q and r in the above families of solution, the physical quantities u and v might possess physically significant rich structures.
In this article, the alternative (G’/G)-expansion method has been modified by introducing the generalized Riccati equation mapping and obtain abundant exact traveling wave solutions of the (1+1)-dimensional DSW equation with the help of symbolic computation. It is important to point out that the obtained solutions have not been reported in the previous literature. The new type of traveling wave solutions found in this article might have significant impact on future research. We assured the correctness of our solutions by putting them back into the original Equations (6) and (7). This article is only an imploring work and we look forward the modified alternative (G’/G)-expansion method may be applicable to other kinds of NLEEs in mathematical physics. The extension of the method proposed in this paper to solve NLEEs with variable coefficients deserves further investigations.
The authors would like to express their sincere thanks to the School of Mathematical Sciences, Universiti Sains Malaysia (USM), Malaysia for providing the symbolic computation software to facilitate the huge calculation of this article.
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