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

M Ali Akbar,Syed Tauseef Mohyud-Din,Norhashidah Hj Mohd Ali

doi:10.1186/2193-1801-2-327

M Ali Akbar, Syed Tauseef Mohyud-Din + Show 1 more

Open Access

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

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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.

Highlights

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 active areas of research
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 new type of traveling wave solutions found in this article might have significant impact on future research

Summary

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 active areas of research. 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. Zayed (2009b) presented a new approach of the (G’/G)-expansion method where G(ξ) satisfies the Jacobi elliptic equation [G′(ξ)]2 = e2G4(ξ) + e1G2(ξ) + e0, e2, e1, e0 are arbitrary constants, and obtained new exact solutions. 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.

Δ2 ðΔ ξÞ csc ðΔ ξÞ þ 2 Δ cos ðΔ

Δ2 cscðΔ ξÞfp

Ω A2 þ B2 þ 2 A Ω

Conclusion