Dynamical characteristics of internal gravity waves in fluid layers with the emergence of mixed solitons

The Kakutani–Matsuuchi (KM) equation is a reduced nonlinear model for the propagation of internal gravity waves in stratified fluids with variable density. These waves exhibit rich nonlinear dynamics and can form a variety of solitary wave structures. In this work, the improved modified Sardar sub-equation method (IMSSEM) is applied to derive exact analytical traveling wave solutions of the (1+1)-dimensional KM equation. By reducing the governing nonlinear partial differential equation to an associated ordinary differential equation, the IMSSEM provides a systematic and efficient framework for constructing exact solutions. A wide range of localized wave profiles is obtained, including bright, dark, smooth periodic, periodic bright, anti-bell-shaped, mixed bright–dark, and double-notch dark soliton solutions. These solutions are illustrated through two- and three-dimensional, contour, and density plots, offering clear insight into their structural and propagation characteristics and demonstrating the flexibility of the IMSSEM. In addition, a comprehensive dynamical analysis is conducted using phase portraits, Hamiltonian energy-level curves, bifurcation diagrams, sensitivity analysis, and forced dynamical studies. The results reveal complex behaviors, including transitions from regular motion to chaotic-like dynamics and the emergence of multistability, indicating strong sensitivity to initial conditions. These findings enhance the analytical understanding of nonlinear internal wave dynamics and provide useful insight into nonlinear and chaotic behavior in complex physical and engineering systems.