Role of shallow water waves generated by modified Camassa-Holm equation: A comparative analysis for traveling wave solutions

Abstract

There is no denying fact that harmonic crystals, cold plasma or liquids and compressible fluids are usually dependent of acoustic-gravity waves, acoustic waves, hydromagnetic waves, surface waves with long wavelength and few others. In this context, the exact solutions of the modified Camassa-Holm equation have been successfully constructed on the basis of comparative analysis of (G'/G - 1/G)and (1/G')-expansion methods. The (G'/G - 1/G)and (1/G')-expansion methods have been proved to be powerful and systematic tool for obtaining the analytical solutions of nonlinear partial differential equations so called modified Camassa-Holm equation. The solutions investigated via (G'/G - 1/G) and (1/G')-expansion methods have remarkably generated trigonometric, hyperbolic, complex hyperbolic and rational traveling wave solutions. For the sake of different traveling wave solutions, we depicted 3-dimensional, 2-dimensional and contour graphs subject to the specific values of the parameters involved in the governing equation. Two methods, which are important instruments in generating traveling wave solutions in mathematics, were compared both qualitatively and quantitatively. In addition, advantages and disadvantages of both methods are discussed and their advantages and disadvantages are revealed.