Computational Fractional Dynamical Systems

25Exp(−φ(ξ))‐Expansion Method

25.1 Introduction

The exp(−φ(ξ))‐expansion method is used for finding solitary and periodic solutions to nonlinear partial differential equations. This method was utilized by Akbulut et al. (Akbulut et al. 2017) to obtain the solution of the Zakharov Kuznetsov–Benjamin Bona Mahony (ZK‐BBM) equation and ill‐posed Boussinesq equation. Hafez (2016) implemented this method to construct the new exact traveling wave solutions of the (3 + 1)‐dimensional coupled Klein–Gordon–Zakharov equation arising in mathematical physics and engineering. With the help of the exp(−φ(ξ))‐expansion method, Islam (2015) examined traveling wave solutions of the Benney–Luke problem. Hafez and Akbar (2015) obtained new explicit and exact traveling wave solutions of the (1 + 1)‐dimensional nonlinear Klein–Gordon–Zakharov equation, which describes the interaction of the Langmuir wave and the ion‐acoustic wave in high‐frequency plasma. The exact traveling wave solutions of the Zhiber–Shabat equation have been studied by Hafez et al. (2014). Alam et al. (2015a) found the exact traveling wave solutions to the (3 + 1)‐dimensional modified Korteweg de Vries (mKdV)–ZK and the (2 + 1)‐dimensional Burgers equations using this approach. The solutions to the space–time fractional nonlinear Whitham–Broer–Kaup and generalized nonlinear Hirota–Satsuma coupled Korteweg de Vries (KdV) equations were obtained using this technique by Moussa and Alhakim (2020). Alam et al. (2015b) used this approach ...