Formation of advanced soliton dynamics through the M-fractional regularized long-wave equation

Area of Science:

• Nonlinear Dynamics
• Applied Mathematics
• Fluid Mechanics

Background:

• The time-fractional regularized long-wave (Tf-RLW) model describes solitary wave propagation in shallow water, plasma, and ion-acoustic waves.
• Fractional derivatives are essential for modeling complex phenomena in science and engineering, including tsunami wave mitigation.

Purpose of the Study:

• To compute exact solutions for the Tf-RLW model using novel analytical techniques.
• To analyze and justify the obtained wave solutions in comparison to existing ones.
• To investigate the stability of the derived solutions.

Main Methods:

• Application of the modified F-expansion method.
• Utilization of the extended modified F-expansion method.
• Employing the unified method for solving nonlinear fractional differential equations.

Main Results:

• The study successfully derived various exact wave solutions, including bright-dark bell waves, periodic rogue waves, singular kink waves, and kinky-periodic bell waves.
• Stability analysis of the obtained solutions was performed.
• Solutions were validated using Maple 18.

Conclusions:

• The employed analytical techniques are effective for simplifying the Tf-RLW model and obtaining exact solutions.
• The findings contribute to understanding complex wave phenomena in fluid dynamics and plasma physics.
• The research validates the utility of fractional calculus in modeling real-world wave behaviors.