Extended Benjamin–Ono equation describing internal solitary waves in stratified fluids

This work explores the propagation of internal solitary waves (ISWs) in stratified fluids. By employing multi-scale analysis and a perturbation expansion, we derive an extended Benjamin–Ono (BO) equation describing the complex dynamical behavior of ISWs. Subsequently, the modified Kudryashov method and the (G′G)-expansion method are applied to obtain a variety of exact solutions, including bright, dark, kink, and anti-kink solitons, as well as singular periodic solutions expressed in terms of hyperbolic, trigonometric, and rational functions. Furthermore, we propose a novel physics-informed neural network (PINN) architecture for solving this model. Numerical results demonstrate that our PINN effectively simulates diverse soliton dynamics and accurately predicts their spatiotemporal evolution. Compared with analytical methods, this framework exhibits robust generalization ability and can adapt to different types of solutions by adjusting the equation constraints. Notably, by considering the external periodic forcing term, we also investigate the chaotic behavior of this model in detail. These findings offer new insights into the complex dynamics of ISWs and contribute to a broader understanding of nonlinear wave phenomena.