Bifurcation analysis and new waveforms and dynamical behaviors of the modified regularized long-wave equation

The present article examines the modified regularized long-wave equation. The Sardar sub-equation approach is employed to conduct the investigation. Periodic, kink, and solitary solitons are among the many exact solutions that are obtained. Numerous physical structures found in analytical solutions are visually examined to show how they change dynamically in 3D, 2D and contour plots. The accuracy of the obtained findings is further validated by substituting the solutions into the governing equation using Maple software. The Galilean transformation is applied to derive the dynamical system of the proposed equation. The aforementioned model is then analyzed for bifurcation, chaotic, and sensitivity behaviors based on the principles of planar dynamical systems. The dynamics of the model exhibit periodic, quasi-periodic, and chaotic motions. The results demonstrate the effectiveness, efficiency, and simplicity of the integration methods used, suggesting that these methods can also be applied to investigate more complex nonlinear models in modern scientific and engineering applications. The originality of this study lies in its comprehensive analysis of the governmental model, revealing its intricate dynamics and diverse wave structures. Through the combination of bifurcation analysis, chaotic behavior, stability analysis, and waveform characteristics, this study also contributes to our understanding of nonlinear wave properties in shallow water • The present article examines the modified regularized long-wave equation; • The Sardar sub-equation approach is employed to conduct the investigation • Periodic, kink, and solitary solitons are among the many exact solutions that are obtained; • Numerous physical structures found in analytical solutions are visually examined to show how they change dynamically in 3D, 2D and contour plots; • A perturbation term is introduced to define quasi periodic and quasi periodic-chaotic patterns.