Diverse approaches to search for solitary wave solutions of the fractional modified Camassa–Holm equation

Abstract

In this study, an integrable dispersive modified Camassa–Holm equation is considered with the essence of fractional beta derivative. The aforesaid equation is a shallow water equation and a bi-Hamiltonian having an associated isospectral problem of second order. Three diverse techniques namely the extended Jacobi's elliptic function expansion, the new version of Kudryashov and the Expa function methods are enforced. A variety of complex solitary wave solutions including, Jacobi's elliptic function solutions, bright and dark solitons and many other analytical solutions are developed. The obtained results are explicated graphically depending upon the physical and fractional parameters. These results may also be used to illuminate the significance of applied methods to many other related non-linear physical phenomena.