On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques

An analytical study of the Rosenau equation, which is crucial for analyzing wave phenomena in a variety of physical systems where nonlinear dynamics and dispersion are significant, includes fluid dynamics, plasma physics, and materials science. This equation was proposed to explain the dynamic behavior of dense discrete systems. We employed the Sardar subequation approach and the generalized Riccati equation mapping method to derive analytical solutions. By applying the fractional wave transformation and the conformal fractional derivative, we were able to obtain these solutions. Using these techniques, we obtained a variety of solutions in the form of exponential, trigonometric, mixed hyperbolic, rational, and hyperbolic functions. The solutions include numerous solitary wave solutions, as well as bright and dark soliton solutions. By varying the parameter values, the analytical soliton solutions are further visualized through 2D, 3D, and contour plots using Mathematica 13.0.