In this section, the modified alternative (G’/G)-expansion method is employed to construct some new traveling wave solutions of the (1+1)-dimensional Drinfel’d-Sokolov-Wilson (DSW) equation which is very important nonlinear evolution equation in mathematical physics and engineering and have been paid attention by many researchers. Some exact solutions of the DSW equation were found in the literature. In general, the solutions of the DSW equation have been obtained by means of an ansatz method. Included in the methods are the elliptic-function (Chen & Zhang2003; Liu et al.2005), Exp-function (He et al.2010), Darboux transformation (Guo & Wu2010), improved F-expansion (Zha & Zhi2008), Variational iteration (Zhang2011) and Adomian’s decomposition (Inc2006). 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:
(6)
Now, we use the wave transformation Equation (2) into Equations (6) and (7), which yield:
(8)
(9)
According to step 3, the solution of Equations (8) and (9) can be expressed by a polynomial in (G’/G) as follows:
(10)
and
(11)
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
(12)
(13)
By means of Equation (5), Equations (12) and (13) can be rewritten respectively as,
(14)
and
(15)
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:
(16)
Solving the over-determined set of algebraic equations by using the symbolic computation software, such as, Maple, we obtain
(17)
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).
Family 1
When p
2 − 4 q r < 0 and pq≠0 (or r q≠0), the periodic form solutions of Equations (6) and (7) are:
where and b
1, p, q, r are arbitrary constants.
where A and B are two non-zero real constants satisfies the condition A
2 − B
2 > 0.
Family 2
When p
2 − 4 q r > 0 and pq≠0 (or rq≠0), the soliton and soliton-like solutions of Equations (6) and (7) are:
where and b
1, p, q, r are arbitrary constants.
where A and B are two non-zero real constants and satisfies the condition B
2 − A
2 > 0.
Family 3
When r=0 and pq≠0, the solutions of Equations (6) and (7) are:
Family 4
When q≠0 and r=p=0, the solutions of Equations (6) and (7) are:
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.