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

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 Exp(a) 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.

Automated summary

This study investigates an integrable dispersive modified Camassa-Holm equation with fractional beta derivative, utilizing techniques such as extended Jacobi's elliptic function expansion, the new version of Kudryashov, and the Exp(a) function methods. Various complex solitary wave solutions are developed, including Jacobi's elliptic function solutions, bright and dark solitons, and others, with graphical explanations based on physical and fractional parameters. These results can illuminate the significance of applied methods in studying related nonlinear physical phenomena.