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

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.

Exact solutions of the nonlinear differential—difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete (G′/G)-expansion method

“…with u 0 = 1/ √ LC 0 . Note that the solution equation (41) has been obtained by Remoissenet [46] after applying the longwavelength approximation. He called it a voltage-pulse soliton.…”

“…(27) and (28) with a bright-like soliton profile, while the solutions described through Eqs. (41) and (42) have been plotted in Fig. 3.…”

“…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 a considerable number of researchers. Akbar et al [41] proposed the modified alternative (G /G)-expansion method to find traveling wave solutions of Drinfel'd-Sokolov-Wilson equation, Shakeel et al [42] applied an improved (G /G)-expansion method for Burger's, Zakharov-Kuznetsov (ZK) and Boussinesq equations. The new approach of generalized (G /G)-expansion method has been applied in Ref.…”

Exact Solutions of the Two Dimensional KdV-Burger Equation by Generalized Kudryashov Method

“…Therefore, different approximation methods have been proposed and developed. Examples of these exact solution methods are exp function method (He and Wu, 2006;Ravi et al, 2017), tanh function method (Malfliet and Hereman, 1996;Malfliet, 2004),   / GG  -expansion method (Akbar et al, 2013;Shakeel and Mohyud-Din, 2015), trial equation method (Liu, 2006;Liu, 2010;Gurefe et al, 2011;, improved   / GG  -expansion method (Zhang et al, 2010;Guo and Zhou, 2010), extended trial equation method (Pandir et al, 2012;Gurefe et al, 2013), multiple extended trial equation method , Jakobi elliptical function method (Fu, 2001;Shen and Pan, 2003), Kudryashov method (Kudryashov, 2012;Ryabov et al, 2011;Lee and Sakthivel, 2013), modified Kudryashov method (Pandir, 2014;Tandogan et al, 2013), a new version of the generalized F-expansion method (Pandir and Turhan, 2014), a new type of the generalized Fexpansion method (Pandir, 2017). Nikolay A. Kudryashov (Kudryashov, 2012) proposed an effective method to obtain the exact solutions of the nonlinear differential equations.…”