SpringerPlus

Impact Factor 0.982

SpringerPlus Cover Image
Research
Open Access

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

  • M Ali Akbar1, 2Email author,
  • Norhashidah Hj Mohd Ali2 and
  • Syed Tauseef Mohyud-Din3
SpringerPlus20132:327

https://doi.org/10.1186/2193-1801-2-327

©  Akbar et al.; licensee Springer. 2013

Received: 1 March 2013

Accepted: 20 June 2013

Published: 18 July 2013

Abstract

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.

Keywords

(G’/G)-expansion method Travelling wave solutions DSW equation Nonlinear evolution equations 02.30.Jr 05.45.Yv 02.30.Ik

Introduction

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.

The method

Suppose the general nonlinear partial differential equation,
P u , u t , u x , u t t , u t x , u x x , ⋯ = 0
(1)

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
u x , t = u ξ , ξ = x − Vt
(2)
where V is the speed of the traveling wave, permits us to transform the Equation (1) into an ODE:
Q u , u ′ , u ″ , ⋯ = 0
(3)

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:
u ξ = ∑ n = 0 m a n G ′ G n , a m ≠ 0
(4)
where G’/G(ξ) satisfies the generalized Riccati equation,
G ′ = r + p G + q G 2 ,
(5)

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,
G 1 = 1 2 q − p + 4 q r − p 2 tan 1 2 4 q r − p 2 ξ ,
G 2 = − 1 2 q p + 4 q r − p 2 cot 1 2 4 q r − p 2 ξ ,
G 3 = 1 2 q − p + 4 q r − p 2 tan 4 q r − p 2 ξ ± sec 4 q r − p 2 ξ ,
G 4 = − 1 2 q p + 4 q r − p 2 cot 4 q r − p 2 ξ ± csc 4 q r − p 2 ξ ,
G 5 = 1 4 q − 2 p + 4 q r − p 2 tan 1 4 4 q r − p 2 ξ − cot 1 4 4 q r − p 2 ξ ,
G 6 = 1 2 q − p + A 2 − B 2 4 q r − p 2 − A 4 q r − p 2 cos 4 q r − p 2 ξ A sin 4 q r − p 2 ξ + B ,
G 7 = 1 2 q − p + A 2 − B 2 4 q r − p 2 + A 4 q r − p 2 cos 4 q r − p 2 ξ A sin 4 q r − p 2 ξ + B ,
where A and B are two non-zero real constants and satisfies the condition A 2 − B 2 > 0.
G 8 = − 2 r cos 1 2 4 q r − p 2 ξ 4 q r − p 2 sin 1 2 4 q r − p 2 ξ + p cos 1 2 4 q r − p 2 ξ ,
G 9 = 2 r sin 1 2 4 q r − p 2 ξ − p sin 1 2 4 q r − p 2 ξ + 4 q r − p 2 cos 1 2 4 q r − p 2 ξ ,
G 10 = − 2 r cos 4 q r − p 2 ξ 4 q r − p 2 sin 4 q r − p 2 ξ + p cos 4 q r − p 2 ξ ± 4 q r − p 2 ,
G 11 = 2 r sin 4 q r − p 2 ξ − p sin 4 q r − p 2 ξ + 4 q r − p 2 cos 4 q r − p 2 ξ ± 4 q r − p 2 ,
G 12 = 4 r sin 1 4 4 q r − p 2 ξ cos 1 4 4 q r − p 2 ξ − 2 p sin 1 4 4 q r − p 2 ξ cos 1 4 4 q r − p 2 ξ + 2 4 q r − p 2 cos 2 1 4 4 q r − p 2 ξ − 4 q r − p 2 .
Family 1: When p 2 − 4 q r > 0 and pq≠0 (or r q≠0), the solutions of Equation (5) are,
G 13 = − 1 2 q p + p 2 − 4 q r tanh 1 2 p 2 − 4 q r ξ ,
G 14 = − 1 2 q p + p 2 − 4 q r coth 1 2 p 2 − 4 q r ξ ,
G 15 = − 1 2 q p + p 2 − 4 q r tanh p 2 − 4 q r ξ ± i sec h p 2 − 4 q r ξ ,
G 16 = − 1 2 q p + p 2 − 4 q r coth p 2 − 4 q r ξ ± csc h p 2 − 4 q r ξ ,
G 17 = − 1 4 q 2 p + p 2 − 4 q r tanh 1 4 p 2 − 4 q r ξ + coth 1 4 p 2 − 4 q r ξ ,
G 18 = 1 2 q − p + A 2 + B 2 p 2 − 4 q r − A p 2 − 4 q r cosh p 2 − 4 q r ξ A sinh p 2 − 4 q r ξ + B ,
G 19 = 1 2 q − p − B 2 − A 2 p 2 − 4 q r + A p 2 − 4 q r cosh p 2 − 4 q r ξ A sinh p 2 − 4 q r ξ + B ,
where A and B are two non-zero real constants and satisfies the condition B 2 − A 2 > 0.
G 20 = 2 r cosh 1 2 p 2 − 4 q r ξ p 2 − 4 q r sinh 1 2 p 2 − 4 q r ξ − p cosh 1 2 p 2 − 4 q r ξ ,
G 21 = 2 r sinh 1 2 p 2 − 4 q r ξ p 2 − 4 q r cosh 1 2 p 2 − 4 q r ξ − p sinh 1 2 p 2 − 4 q r ξ ,
G 22 = 2 r cosh p 2 − 4 q r ξ p 2 − 4 q r sinh p 2 − 4 q r ξ − p cosh p 2 − 4 q r ξ ± i p 2 − 4 q r ,
G 23 = 2 r sinh p 2 − 4 q r ξ − p sinh p 2 − 4 q r ξ + p 2 − 4 q r cosh p 2 − 4 q r ξ ± p 2 − 4 q r ,
G 24 = 4 r sinh 1 4 p 2 − 4 q r ξ cosh 1 4 p 2 − 4 q r ξ − 2 p sinh 1 4 p 2 − 4 q r ξ cosh 1 4 p 2 − 4 q r ξ + 2 p 2 − 4 q r cosh 2 1 4 p 2 − 4 q r ξ − p 2 − 4 q r .
Family 2: When r=0 and pq≠0, the solutions of Equation (5) are,
G 25 = − p d q d + cosh p ξ − sinh p ξ ,
G 26 = − p cosh p ξ + sinh p ξ q d + cosh p ξ + sinh p ξ ,

where d is an arbitrary constant.

Family 3: When q≠0 and r=p=0, the solution of Equation (5) is,
G 27 = − 1 q ξ + c 1 ,

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).

Some new traveling wave solutions of the DSW equation

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:
2 v x x x + 2 u v x + u x v − v t = 0
(6)
3 v v x − u t = 0 .
(7)
Now, we use the wave transformation Equation (2) into Equations (6) and (7), which yield:
2 v ‴ + 2 u v ′ + v u ′ + V v ′ = 0 ,
(8)
3 v v ′ + V u ′ = 0 .
(9)
According to step 3, the solution of Equations (8) and (9) can be expressed by a polynomial in (G’/G) as follows:
u ξ = a 0 + a 1 G ′ G + a 2 G ′ G 2 + ⋯ + a m G ′ G m , a m ≠ 0
(10)
and
v ξ = b 0 + b 1 G ′ G + b 2 G ′ G 2 + ⋯ + b n G ′ G n b n ≠ 0
(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
u ξ = a 0 + a 1 G ′ G + a 2 G ′ G 2 , a 2 ≠ 0
(12)
v ξ = b 0 + b 1 G ′ G , b 1 ≠ 0
(13)
By means of Equation (5), Equations (12) and (13) can be rewritten respectively as,
u ξ = a 0 + a 1 p + r G − 1 + q G + a 2 p + r G − 1 + q G 2
(14)
and
v ξ = b 0 + b 1 p + r G − 1 + q G .
(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:
24 b 1 p + 3 a 1 b 1 + 2 a 2 b 0 + 12 a 2 b 1 p = 0 , 2 a 2 V + 3 b 1 2 = 0 , a 2 b 1 + 3 b 1 = 0 ,
a 1 b 0 + 6 a 1 b 1 p + V b 1 + 4 a 2 b 0 p + 16 b 1 q r + 2 a 0 b 1 + 8 a 2 b 1 q r + 12 a 2 b 1 p 2 + 14 b 1 p 2 = 0 ,
3 a 1 b 1 r q + 4 a 2 b 1 p 3 + V b 1 p + 3 a 1 b 1 p 2 + 2 a 2 b 0 p 2 + 2 a 0 b 1 p + 2 b 1 p 3 + a 1 b 0 p + 2 a 2 b 0 r q + 16 b 1 p q r + 12 a 2 p q r = 0 ,
V b 1 p + a 1 b 0 p + 2 a 0 b 1 p + 12 a 2 b 1 p q r + 4 a 2 b 1 p 3 + 3 a 1 b 1 p 2 + 2 b 1 p 3 + 2 a 2 b 0 p 2 + 2 a 2 b 0 r q + 3 a 1 b 1 q r + 16 b 1 p q r = 0 ,
a 1 b 0 + 8 a 2 b 1 r q + 6 a 1 b 1 p + 14 b 1 p 2 + 4 a 2 b 0 p + 16 b 1 q r + 12 a 2 b 1 p 2 + 2 a 0 b 1 + V b 1 = 0 ,
3 a 1 b 1 + 24 b 1 p + 2 a 2 b 0 + 12 a 2 b 1 p = 0 , V a 1 + 4 a 2 Vp + 3 b 0 b 1 + 6 b 1 2 p = 0 ,
(16)
3 b 0 b 1 p + 3 b 1 2 r q + 3 b 1 2 p 2 + a 1 p V + 2 a 2 p 2 V + 2 a 2 r qV = 0 , 3 b 0 b 1 + 4 a 2 p V + 6 b 1 2 p + a 1 V = 0 ,
3 b 0 b 1 p + 3 b 1 2 p 2 + a 1 p V + 2 a 2 p 2 V + 3 b 1 2 q r + 2 a 2 q r V = 0 .
Solving the over-determined set of algebraic equations by using the symbolic computation software, such as, Maple, we obtain
a 2 = − 3 , a 1 = 3 p , a 0 = − p 2 4 − b 1 2 4 + 4 q r , b 1 = b 1 , b 0 = − 1 2 b 1 p , V = 1 2 b 1 2
(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:
u 1 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Δ 2 sec 2 Δ ξ − p + 2 Δ tan Δ ξ − 3 2 Δ 2 sec 2 Δ ξ − p + 2 Δ tan Δ ξ 2 ,
v 1 = − 1 2 b 1 p + b 1 2 Δ 2 sec 2 Δ ξ − p + 2 Δ tan Δ ξ ,
where Δ = 1 2 4 q r − p 2 , ξ = x − 1 2 b 1 2 t and b 1, p, q, r are arbitrary constants.
u 2 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Δ 2 csc 2 Δ ξ p + 2 Δ cot Δ ξ − 3 2 Δ 2 csc 2 Δ ξ p + 2 Δ cot Δ ξ 2 ,
v 2 = − 1 2 b 1 p − b 1 2 Δ 2 csc 2 Δ ξ p + 2 Δ cot Δ ξ ,
u 3 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 4 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ − p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ − 3 4 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ − p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ 2 ,
v 3 = − 1 2 b 1 p + b 1 4 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ − p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ ,
u 4 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 4 Δ 2 csc 2 Δ ξ 1 ± cos 2 Δ ξ p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ − 3 4 Δ 2 csc 2 Δ ξ 1 ± cos 2 Δ ξ p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ 2 ,
v 4 = − 1 2 b 1 p − b 1 4 Δ 2 csc 2 Δ ξ 1 ± cos 2 Δ ξ p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ ,
u 5 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Δ 2 csc Δ ξ p sin Δ ξ + 2 Δ cos Δ ξ − 3 2 Δ 2 csc Δ ξ p sin Δ ξ + 2 Δ cos Δ ξ 2 ,
v 5 = − 1 2 b 1 p − b 1 2 Δ 2 csc Δ ξ p sin Δ ξ + 2 Δ cos Δ ξ ,
u 6 = − 3 p 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ − B sin 2 Δ ξ − A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 − B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ + 2 A Δ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 − 3 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ − B sin 2 Δ ξ − A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 0− B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ + 2 AΔ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 2 − p 2 4 − b 1 2 4 + 4 q r ,
v 6 = − 1 2 b 1 p − b 1 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ − B sin 2 Δ ξ − A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 − B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ + 2 A Δ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 ,
u 7 = − 3 p 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ + B sin 2 Δ ξ + A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 − B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ − 2 A Δ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 − 3 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ + B sin 2 Δ ξ + A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 − B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ − 2 AΔ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 2 − p 2 4 − b 1 2 4 + 4 q r ,
v 7 = − 1 2 b 1 p − b 1 4 A Δ 2 A 2 − B 2 cos 2 Δ ξ + B sin 2 Δ ξ + A A sin 2 Δ ξ + B A 2 cos 2 2 Δ ξ − A 2 − B 2 − 2 AB sin 2 Δ ξ p A sin 2 Δ ξ − 2 A Δ cos 2 Δ ξ + pB − 2 Δ A 2 − B 2 ,
where A and B are two non-zero real constants satisfies the condition A 2 − B 2 > 0.
u 8 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Δ 2 sec Δ ξ p cos Δ ξ + 2 Δ sin Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ + 4 Δ 2 − 3 2 Δ 2 sec Δ ξ p cos Δ ξ + 2 Δ sin Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ + 4 Δ 2 2 ,
v 8 = − 1 2 b 1 p − b 1 2 Δ 2 sec Δ ξ p cos Δ ξ + 2 Δ sin Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ + 4 Δ 2 ,
u 9 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 − 3 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 2 ,
v 9 = − 1 2 b 1 p + b 1 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 ,
u 10 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ p 2 − 2 r q cos 2 2 Δ ξ + 2 Δ 1 ± sin 2 Δ ξ 2 Δ ± p cos 2 Δ ξ − 3 2 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ p 2 − 2 r q cos 2 2 Δ ξ + 2 Δ 1 ± sin 2 Δ ξ 2 Δ ± p cos 2 Δ ξ 2 ,
v 10 = − 1 2 b 1 p − b 1 2 Δ 2 sec 2 Δ ξ 1 ± sin 2 Δ ξ p cos 2 Δ ξ + 2 Δ sin 2 Δ ξ ± 2 Δ p 2 − 2 r q cos 2 2 Δ ξ + 2 Δ 1 ± sin 2 Δ ξ 2 Δ ± p cos 2 Δ ξ ,
u 11 = − p 2 4 − b 1 2 4 + 4 q r ± 3 p 2 Δ 2 csc 2 Δ ξ − p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ 2 r q − p 2 cos 2 Δ ξ − 2 p Δ sin 2 Δ ξ ± 2 q r − 3 2 Δ 2 csc 2 Δ ξ − p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ 2 r q − p 2 cos 2 Δ ξ − 2 p Δ sin 2 Δ ξ ± 2 q r 2 ,
v 11 = − 1 2 b 1 p ± b 1 2 Δ 2 csc 2 Δ ξ − p sin 2 Δ ξ + 2 Δ cos 2 Δ ξ ± 2 Δ 2 r q − p 2 cos 2 Δ ξ − 2 p Δ sin 2 Δ ξ ± 2 q r ,
u 12 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 − 3 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 2 ,
v 12 = − 1 2 b 1 p + b 1 2 Δ 2 csc Δ ξ p sin Δ ξ − 2 Δ cos Δ ξ 2 p 2 − 2 r q cos 2 Δ ξ + 4 Δ p sin Δ ξ cos Δ ξ − p 2 .

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:
u 13 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Ω 2 sec h 2 Ω ξ p + 2 Ω tanh Ω ξ − 3 2 Ω 2 sec h 2 Ω ξ p + 2 Ω tanh Ω ξ 2 ,
v 13 = − 1 2 b 1 p + b 1 2 Ω 2 sec h 2 Ω ξ p + 2 Ω tanh Ω ξ ,
where Ω = 1 2 p 2 − 4 q r , ξ = x − 1 2 b 1 2 t and b 1, p, q, r are arbitrary constants.
u 14 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Ω 2 csc h 2 Ω ξ p + 2 Δ coth Ω ξ − 3 2 Ω 2 csc h 2 Ω ξ p + 2 Δ coth Ω ξ 2 ,
v 14 = − 1 2 b 1 p − b 1 2 Ω 2 csc h 2 Ω ξ p + 2 Δ coth Ω ξ ,
u 15 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ + 2 Δ sinh 2 Ω ξ ± i 2 Ω − 3 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ + 2 Δ sinh 2 Ω ξ ± i 2 Ω 2 ,
v 15 = − 1 2 b 1 p + b 1 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ + 2 Δ sinh 2 Ω ξ ± i 2 Ω ,
u 16 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ p sinh 2 Ω ξ + 2 Ω cosh 2 Δ ξ ± 2 Ω − 3 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ p sinh 2 Ω ξ + 2 Ω cosh 2 Δ ξ ± 2 Ω 2 ,
v 16 = − 1 2 b 1 p − b 1 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ p sinh 2 Ω ξ + 2 Ω cosh 2 Δ ξ ± 2 Ω ,
u 17 = − p 2 4 − b 1 2 4 + 4 q r − 3 p Ω 2 sec h 2 Ω ξ / 2 2 cosh 2 Ω ξ / 2 − 1 p + Ω tanh Ω ξ / 2 + coth Ω ξ / 2 − 3 Ω 2 sec h 2 Ω ξ / 2 2 cosh 2 Ω ξ / 2 − 1 p + Ω tanh Ω ξ / 2 + coth Ω ξ / 2 2 ,
v 17 = − 1 2 b 1 p − b 1 Ω 2 sec h 2 Ω ξ / 2 2 cosh 2 Ω ξ / 2 − 1 p + Ω tanh Ω ξ / 2 + coth Ω ξ / 2 ,
u 18 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 4 A Ω 2 A − B sinh 2 Ω ξ − A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B − 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ − 3 4 A Ω 2 A − B sinh 2 Ω ξ − A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B − 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ 2 ,
v 18 = − 1 2 b 1 p − b 1 4 A Ω 2 A − B sinh 2 Ω ξ − A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B − 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ ,
u 19 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 4 A Ω 2 A − B sinh 2 Ω ξ + A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B + 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ − 3 4 A Ω 2 A − B sinh 2 Ω ξ + A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B + 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ 2 ,
v 19 = − 1 2 b 1 p − b 1 4 A Ω 2 A − B sinh 2 Ω ξ + A 2 + B 2 cosh 2 Ω ξ A sin 2 Ω ξ + B p A sinh 2 Ω ξ + p B + 2 Ω A 2 + B 2 + 2 A Ω cosh 2 Ω ξ ,
where A and B are two non-zero real constants and satisfies the condition B 2 − A 2 > 0.
u 20 = − p 2 4 − b 1 2 4 + 4 q r − 3 p 2 Ω 2 sec h Ω ξ 2 Ω sinh Ω ξ − p cosh Ω ξ − 3 2 Ω 2 sec h Ω ξ 2 Ω sinh Ω ξ − p cosh Ω ξ 2 ,
v 20 = − 1 2 b 1 p − b 1 2 Ω 2 sec h Ω ξ 2 Ω sinh Ω ξ − p cosh Ω ξ ,
u 21 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ − 3 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ 2 ,
v 21 = − 1 2 b 1 p + b 1 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ ,
u 22 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ − 2 Ω sinh 2 Ω ξ ∓ i 2 Ω − 3 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ − 2 Ω sinh 2 Ω ξ ∓ i 2 Ω 2 ,
v 22 = − 1 2 b 1 p + b 1 4 Ω 2 sec h 2 Ω ξ 1 ∓ i sinh 2 Ω ξ p cosh 2 Ω ξ − 2 Ω sinh 2 Ω ξ ∓ i 2 Ω ,
u 23 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ 2 Ω cosh 2 Ω ξ − p sinh 2 Ω ξ ± 2 Ω − 3 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ 2 Ω cosh 2 Ω ξ − p sinh 2 Ω ξ ± 2 Ω 2 ,
v 23 = − 1 2 b 1 p + b 1 4 Ω 2 csc h 2 Ω ξ 1 ± cosh 2 Ω ξ 2 Ω cosh 2 Ω ξ − p sinh 2 Ω ξ ± 2 Ω ,
u 24 = − p 2 4 − b 1 2 4 + 4 q r + 3 p 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ − 3 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ 2 ,
v 24 = − 1 2 b 1 p + b 1 2 Ω 2 csc h Ω ξ 2 Ω cosh Ω ξ − p sinh Ω ξ .

Family 3

When r=0 and pq≠0, the solutions of Equations (6) and (7) are:
u 25 = − p 2 4 − b 1 2 4 + 4 q r + 3 p p cosh p ξ − sinh p ξ d + cosh p ξ − sinh p ξ − 3 p cosh p ξ − sinh p ξ d + cosh p ξ − sinh p ξ 2 ,
v 25 = − 1 2 b 1 p + b 1 p cosh p ξ − sinh p ξ d + cosh p ξ − sinh p ξ ,
u 26 = − p 2 4 − b 1 2 4 + 4 q r + 3 p p d d + cosh p ξ + sinh p ξ − 3 p d d + cosh p ξ + sinh p ξ 2 ,
v 26 = − 1 2 b 1 p + b 1 p d d + cosh p ξ + sinh p ξ .

Family 4

When q≠0 and r=p=0, the solutions of Equations (6) and (7) are:
u 27 = − p 2 4 − b 1 2 4 + 4 q r − 3 p q q ξ + c 1 − 3 q q ξ + c 1 2 ,
v 27 = − 1 2 b 1 p − b 1 q q ξ + c 1

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.

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 and5).
Figure 1

Solitons corresponding to solutions u 1 and v 1 for p = q =2, r =3 and b 1 =1.

Figure 2

Solitons corresponding to solutions u 5 and v 5 for p = q =1, r =2 and b 1 =1.

Figure 3

Solitons corresponding to solutions u 13 and v 13 for p =3, q =2, r =1 and b 1 =1.

Figure 4

Solitons corresponding to solutions u 24 and v 24 for p =3, q =2, r =1 and b 1 =5.

Figure 5

Solitons corresponding to solutions u 27 and v 27 for p =0, q =1, r =0 and b 1 =5 and c 1 =1.

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.

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
Department of Applied Mathematics, University of Rajshahi
(2)
School of Mathematical Sciences, Universiti Sains Malaysia
(3)
Department of Mathematics, HITEC University

References

  1. Abazari R: The ( G ’/ G )-expansion method for Tziteica type nonlinear evolution equations. Math Comput Modelling 2010, 52: 1834-1845. 10.1016/j.mcm.2010.07.013View ArticleGoogle Scholar
  2. Abdou MA, Soliman AA, Basyony ST: New application of exp-function method for improved Boussinesq equation. Phys Lett A 2007, 369: 469-475. 10.1016/j.physleta.2007.05.039View ArticleGoogle Scholar
  3. Akbar MA, Ali NHM: Exp-function method for duffing equation and New solutions of (2+1) dimensional dispersive long wave equations. Prog Appl Math 2011, 1(2):30-42.Google Scholar
  4. Akbar MA, Ali NHM, Zayed EME: Abundant exact traveling wave solutions of the generalized Bretherton equation via the improved (G'/G)-expansion method. Commun. Theor. Phys. 2012, 57: 173-178. 10.1088/0253-6102/57/2/01View ArticleGoogle Scholar
  5. Ali AT: New generalized Jacobi elliptic function rational expansion method. J Comput Appl Math 2011, 235: 4117-4127. 10.1016/j.cam.2011.03.002View ArticleGoogle Scholar
  6. Bekir A: Application of the ( G ’/ G )-expansion method for nonlinear evolution equations. Phys Lett A 2008, 372: 3400-3406. 10.1016/j.physleta.2008.01.057View ArticleGoogle Scholar
  7. Chen HT, Zhang HQ: Improved Jacobin elliptic function method and its applications. Chaos Solitons and Fract 2003, 15: 585-591. 10.1016/S0960-0779(02)00147-9View ArticleGoogle Scholar
  8. Gardner CS, Greener JM, Kruskal MD, et al.: Phys Rev Lett. 1967, 19: 1095-1099. 10.1103/PhysRevLett.19.1095View ArticleGoogle Scholar
  9. Guo GX, Wu LH: Darboux transformation and explicit solutions for Drinfel’d–Sokolov–Wilson Equation. Commun Theor Phys 2010, 53: 1090-1096. 10.1088/0253-6102/53/6/20View ArticleGoogle Scholar
  10. He JH: Asymptotic methods for solitary solutions and compactons. Abstr Appl Anal 2012, 916793: 130. 10.1155/2012/916793Google Scholar
  11. He JH, Wu XH: Exp-function method for nonlinear wave equations. Chaos Solitons Fract 2006, 30: 700-708. 10.1016/j.chaos.2006.03.020View ArticleGoogle Scholar
  12. He YH, Long Y, Li SL: Exact solutions of the Drinfel’d-Sokolov-Wilson Equation using the F-expansion method combined with Exp-function method. Int Math Forum 2010, 5: 3231-3242.Google Scholar
  13. Hirota R: Exact solution of the KdV equation for multiple collisions of solitions. Phys Rev Lett 1971, 27: 1192-1194. 10.1103/PhysRevLett.27.1192View ArticleGoogle Scholar
  14. Inc M: On numerical doubly periodic wave solutions of the coupled Drinfel’d–Sokolov–Wilson equation by the decomposition method. Appl Math Comput 2006, 172: 421-430. 10.1016/j.amc.2005.02.012View ArticleGoogle Scholar
  15. Jafari MA, Aminataei A: Improvement of the homotopy perturbation method for solving nonlinear diffusion equations. Phys Scr 2010, 82: 5. 015002View ArticleGoogle Scholar
  16. Kol GR, Tabi CB: Application of the ( G ’/ G )-expansion method to nonlinear blood flow in large vessels. Phys. Scr. 2011, 83: 6. 045803View ArticleGoogle Scholar
  17. Liu S, Fu Z, Liu S, Zhao Q: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001, 289: 69-74. 10.1016/S0375-9601(01)00580-1View ArticleGoogle Scholar
  18. Liu S, Fu Z, Liu S: Periodic solutions for a class of coupled nonlinear partial differential equations. Phys Lett A 2005, 336: 175-179. 10.1016/j.physleta.2005.01.025View ArticleGoogle Scholar
  19. Liu X, Tian L, Wu Y: Exact solutions of the generalized Benjamin-Bona-Mahony equation. Math. Prob. Engr. 2010, 796398: 5. 10.1155/2010/796398Google Scholar
  20. Malfliet M: Solitary wave solutions of nonlinear wave equations. Am J Phys 1992, 60: 650-654. 10.1119/1.17120View ArticleGoogle Scholar
  21. Marinca V, Herişanu N: Explicit and exact solutions to cubic Duffing and double-well Duffing equations. Math Comp Model 2011, 53: 604-609. 10.1016/j.mcm.2010.09.011View ArticleGoogle Scholar
  22. Naher H, Abdullah FA, Akbar MA: The ( G ’/ G )-expansion method for abundant travelling wave solutions of Caudrey-Dodd-Gibbon equation. Math Prob Engr 2011. Article ID 218216, 11 pages 10.1155/2011/218216Google Scholar
  23. Naher H, Abdullah FA, Akbar MA: New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. J. Appl. Math. 2012. Article ID 575387 14 pages 10.1155/2012/575387Google Scholar
  24. Rogers C, Shadwick WF: Backlund Transformations. New York: Academic Press; 1982.Google Scholar
  25. Wang ML: Exact solutions for a compound KdV-Burgers equation. Phys Lett A 1996, 213: 279-287. 10.1016/0375-9601(96)00103-XView ArticleGoogle Scholar
  26. Wang ML, Li X, Zhang J: The ( G ’/ G )-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys Lett A 2008, 372: 417-423. 10.1016/j.physleta.2007.07.051View ArticleGoogle Scholar
  27. Wazwaz MA: Partial Differential Equations and Solitary Waves Theory. London, New York: Springer Dordrecht Heidelberg; 2009.View ArticleGoogle Scholar
  28. Yang XJ: Advanced local fractional calculus and its applications. World Science Publisher; 2012.Google Scholar
  29. Zayed EME: The ( G ’/ G )-expansion method and its applications to some nonlinear evolution equations in the mathematical physics. J Appl Math Comput 2009, 30: 89-103. 10.1007/s12190-008-0159-8View ArticleGoogle Scholar
  30. Zayed EME: New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized ( G ’/ G )-expansion method. J Phys A: Math Theor 2009, 42: 195202-195214. 10.1088/1751-8113/42/19/195202View ArticleGoogle Scholar
  31. Zayed EME: The ( G ’/ G )-expansion method combined with the Riccati equation for finding exact solutions of nonlinear PDEs. J Appl Math Inform 2011, 29(1–2):351-367.Google Scholar
  32. Zayed EME, Gepreel KA: The ( G ’/ G )-expansion method for finding traveling wave solutions of nonlinear PDEs in mathematical physics. J Math Phys 2009, 50: 013502-013513. 10.1063/1.3033750View ArticleGoogle Scholar
  33. Zha XQ, Zhi HY: An Improved F-Expansion Method and its Application to Coupled Drinfel’d–Sokolov–Wilson Equation. Commun Theor Phys 2008, 50: 309-314. 10.1088/0253-6102/50/2/05View ArticleGoogle Scholar
  34. Zhang WM: Solitary solutions and singular periodic solutions of the Drinfeld-Sokolov-Wilson Equation by variational approach. Appl Math Sci 2011, 5(38):1887-1894.Google Scholar
  35. Zhang S, Tong J, Wang W: A generalized ( G ’/ G )-expansion method for the mKdV equation with variable coefficients. Phys Lett A 2008, 372: 2254-2257. 10.1016/j.physleta.2007.11.026View ArticleGoogle Scholar
  36. Zhang J, Wei X, Lu Y: A generalized ( G ’/ G )-expansion method and its applications. Phys Lett A 2008, 372: 3653-3658. 10.1016/j.physleta.2008.02.027View ArticleGoogle Scholar
  37. Zhang J, Jiang F, Zhao X: An improved ( G ’/ G )-expansion method for solving nonlinear evolution equations. Int J Comput Math 2010, 87: 1716-1725. 10.1080/00207160802450166View ArticleGoogle Scholar
  38. Zhaqilao : Darboux transformation and multi-soliton solutions for some (2+1)-dimensional nonlinear equations. Phys Scr 2010, 82: 5. 035001View ArticleGoogle Scholar
  39. Zhou YB, Wang ML, Wang YM: Periodic wave solutions to coupled KdV equations with variable coefficients. Phys Lett A 2003, 308: 31-36. 10.1016/S0375-9601(02)01775-9View ArticleGoogle Scholar
  40. Zhu S: The generalized Riccati equation mapping method in nonlinear evolution equation: application to (2+1)-dimensional Boiti-Leon-Pempinelle equation. Chaos Soliton and Fract 2008, 37: 1335-1352. 10.1016/j.chaos.2006.10.015View ArticleGoogle Scholar

Copyright

© Akbar et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.