Bifurcation and chaos analysis with solitonic dynamics in the Kakutani-Matsuuchi model

The interplay of dispersion, nonlinearity, and dissipation in internal gravity waves is crucial for understanding phenomena such as wave breaking and energy transport in viscous fluids. The Kakutani-Matsuuchi (KM) model provides a framework for these dynamics, yet a complete characterization of its solution space and stability remains an open challenge. This paper addresses this gap by deriving a rich variety of exact solitary and periodic wave solutions for the KM model using the enhanced modified extended tanh expansion and Sardar sub-equation methods. We report a diverse spectrum of analytical solutions, including previously unreported singular and W-shaped solitary waves, alongside standard bright, dark, and multi-soliton structures. The dynamics and physical relevance of these waves are illustrated through graphical simulations. Furthermore, a comprehensive dynamical analysis reveals the fundamental stability properties of the model. Bifurcation theory is employed to identify the critical parameter thresholds governing transitions between saddle and center equilibria. The system’s route to chaos under parametric forcing is demonstrated through positive Lyapunov exponents and phase space analysis. The model’s structural robustness against minor initial perturbations is also confirmed via sensitivity analysis, providing a complete framework for predicting wave behavior from ordered to chaotic regimes.