The modified alternative (G’/G)-expansion method to nonlinear evolution equation: application to the (1+1)-dimensional Drinfel’d-Sokolov-Wilson equation

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.


Introduction
After the observation of soliton phenomena by John Scott Russell in 1834 (Wazwaz 2009) 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 sechsolutions 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 (Hirota 1971), Backlund transformation (Rogers & Shadwick 1982), improved homotopy perturbation

The method
Suppose the general nonlinear partial differential equation, P u; u t ; u x ; u t t ; u t x ; u x x ; ⋯ ð Þ ¼ 0 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: Step 1: The travelling wave variable ansatz where V is the speed of the traveling wave, permits us to transform the Equation (1) into an ODE: where the superscripts stands for the ordinary derivatives with respect to ξ.
Step 2: Suppose the traveling wave solution of Equation (3) can be expressed by a polynomial in (G'/G) as follows: where G'/G(ξ) satisfies the generalized Riccati equation, where a n (n = 0, 1, 2, ⋯, m), r, p and q are arbitrary constants to be determined later. The generalized Riccati Equation (5) has twenty seven solutions (Zhu, 2008) as follows: Family 1: When p 2 − 4 q r < 0 and pq≠0 (or r q≠0), the solutions of Equation (5) are, where A and B are two non-zero real constants and satisfies the condition A 2 − B 2 > 0.
Family 2: When p 2 − 4 q r > 0 and pq≠0 (or r q≠0), the solutions of Equation (5) are, where A and B are two non-zero real constants and satisfies the condition Family 3: When r=0 and pq≠0, the solutions of Equation (5) are, where d is an arbitrary constant. Family 4: When q≠0 and r=p=0, the solution of Equation (5) is, 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) Adomian's decomposition (Inc 2006). It is to be highlighted that Marinca et. al. (2011) presented quotient trigonometric function expansion method to find explicit and exact solutions to cubic Duffing and double-well Duffing equations. Moreover, a detailed study is made by Yang (2012) on local fractional differential equations and its Applications, Local Fractional Functional Analysis and its Applications along with local fractional variation iteration and local fractional Fourier series methods. He (2012) has also given a comprehensive analysis of Asymptotic methods for solitary solutions and compactons. Inspired and motivated by the ongoing research in this area, we apply the modified alternative (G'/G)-expansion method for searching its new solitary wave solutions. Let us consider the DSW equation: Now, we use the wave transformation Equation (2) into Equations (6) and (7), which yield: According to step 3, the solution of Equations (8) and (9) can be expressed by a polynomial in (G'/G) as follows: and 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.
Therefore, solution Equations (10) and (11) take the form respectively By means of Equation (5), Equations (12) and (13) can be rewritten respectively as, Substituting Equations (14) and (15) into Equations (8) and (9), the left hand sides of these equations are converted into polynomials in G i and G − i , (i = 0, 1, 2, 3, ⋯). Setting each coefficient of these polynomials to zero, we obtain a set of simultaneous algebraic equations for a 0 , a 1 , a 2 , b 0 , b 1 , p, q, r and V as follows: Solving the over-determined set of algebraic equations by using the symbolic computation software, such as, Maple, we obtain where b 1 , p, q and r are arbitrary constants. Now on the basis of the solutions of Equation (5), we obtain some new types of solutions of Equations (6) and (7).
where A and B are two non-zero real constants satisfies the condition A 2 − B 2 > 0. Figure 3 Solitons corresponding to solutions u 13 and v 13 for p=3, q=2, r=1 and b 1 =1.
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.

Graphical presentation
Graph is a powerful tool for communication and describes lucidly the solutions of the problems. Therefore, some graphs of the solutions are given below. The graphs readily have shown the solitary wave form of the solutions (Figures 1, 2, 3, 4 and 5).

Conclusion
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.