Eur. Phys. J. Plus (2020) 135: 726
https://doi.org/10.1140/epjp/s13360-020-00751-8

Regular Article

Soliton solutions to the DNA Peyrard–Bishop equation with beta-derivative via three distinctive approaches

¹ Department of Mathematics, CUI Vehari Campus, Vehari, Pakistan
² Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
³ Department of Mathematics and Statistics, ISP Multan, Multan, Pakistan
⁴ Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawaser, Saudi Arabia

^e n.sooppy@psau.edu.sa

Received: 9 August 2020
Accepted: 3 September 2020
Published online: 14 September 2020

Abstract

In this paper, we explore the DNA dynamic equation arising in the oscillator-chain named as Peyrard–Bishop model for abundant solitary wave solutions. The aforesaid model is studied for the first time via a novel fractional derivative operator. A modified traveling wave transformation with the aforementioned operator is used to transform the space-time fractional Peyrard–Bishop model into an ODE. Then, the three prolific schemes namely the simplest equation method, the Kudryashov method and the modified $(G^{\prime }/G^2)$-expansion method are employed. These methods contribute a variety of exact soliton solutions including the bright, dark and singular solution. Also the obtained solutions are verified through symbolic soft computations. Furthermore, some results are also explained through numerical simulations that show the novelty of our work as compared to the existing literature about the classical Peyrard–Bishop model.