Bifurcation analysis and new waveforms to the first fractional WBBM equation

Area of Science:

• Nonlinear dynamics
• Fluid mechanics
• Mathematical physics

Background:

• Shallow water wave phenomena are often modeled using nonlinear partial differential equations.
• The Wazwaz-Benjamin-Bona-Mahony (WBBM) equation is a significant model in this field.
• Fractional calculus offers a more comprehensive framework for describing complex wave behaviors.

Purpose of the Study:

• To perform bifurcation analysis and identify novel waveforms for the first fractional 3D Wazwaz-Benjamin-Bona-Mahony (WBBM) structure.
• To investigate the linear stability of the model.
• To explore the dynamical system, including chaotic behaviors and sensitivity, of the WBBM equation.

Main Methods:

• Galilean transformation to derive the dynamical system.
• Planar dynamical system principles for bifurcation, chaos, and sensitivity analysis.
• Linear stability technique for model assessment.
• Numerical simulations to visualize wave solutions.

Main Results:

• The study identified periodic, quasi-periodic, and chaotic behaviors within the WBBM model.
• Diverse solitary wave solutions were obtained and visualized, including bright solitons, dark solitons, kink waves, and anti-kink waves.
• The employed integration methods demonstrated effectiveness, brevity, and efficiency.

Conclusions:

• The research provides valuable insights into the complex dynamics and wave structures of the fractional 3D WBBM model.
• The findings advance the understanding of nonlinear wave properties in shallow water environments.
• The methods used are applicable to other complex nonlinear models in science and engineering.