Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation

Abstract

In this study, the analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation, a mathematical model used to describe the propagation of shallow water waves characterized by weak dispersion and nonlinearities, is determined. The (G′/G,1/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({G}^{\\prime}/G,1/G)$$\\end{document}-expansion method, a powerful and reliable technique, is exploited to formulate the soliton solutions for the equation. The process yields a wide range of solutions, including trigonometric, rational, and hyperbolic functions with free parameters. Various original soliton solutions are generated for different parameter values, including bell-shaped, anti-bell-shaped, periodic, compacton, singular bell-shaped, singular periodic and flat kink solitons. To visually comprehend the physical characteristics of the obtained solutions, two- and three-dimensional graphs, as well as contour plots, are plotted. Thorough comparisons with previous results are conducted to ensure the originality of the derived solutions. The insights gained from understanding soliton behavior in shallow water can be helpful in improving tsunami warnings, coastal protection systems, underwater data transmission, submarine cables, and marine sensing networks.