Bifurcation analysis and new waveforms to the first fractional WBBM equation

This research focuses on bifurcation analysis and new waveforms for the first fractional 3D Wazwaz-Benjamin-Bona-Mahony (WBBM) structure, which arises in shallow water waves. The linear stability technique is also employed to assess the stability of the mentioned model. The suggested equation's dynamical system is obtained by applying the Galilean transformation to achieve our goal. Subsequently, bifurcation, chaos, and sensitivity analysis of the mentioned model are conducted by applying the principles of the planar dynamical system. We obtain periodic, quasi-periodic, and chaotic behaviors of the mentioned model. Furthermore, we introduce and delve into diverse solitary wave solutions, encompassing bright soliton, dark soliton, kink wave, periodic waves, and anti-kink waves. These solutions are visually presented through simulations, highlighting their distinct characteristics and existence. The results highlight the effectiveness, brevity, and efficiency of the employed integration methods. They also suggest their applicability to delving into more intricate nonlinear models emerging in modern science and engineering scenarios. The novelty of this research lies in its detailed analysis of the governing model, which provides insights into its complex dynamics and varied wave structures. This study also advances the understanding of nonlinear wave properties in shallow water by combining bifurcation analysis, chaotic behavior, waveform characteristics, and stability assessments.