Painlevé Analysis, Bifurcation Analysis, Chaotic Pattern, and New Exact Solutions to Bidirectional Wave Equations

This study investigates the qualitative behavior of a mathematical model describing bidirectional nonlinear wave propagation in a water channel. The non‐integrability of the model is proved by using Painlevé singularity analysis. By applying a suitable wave transformation, the original system is reduced to a planar dynamical system equivalent to a Hamiltonian system. Within this framework, a bifurcation analysis is performed, leading to a theorem that specifies the conditions for the existence of periodic, solitary, and kink/anti‐kink wave solutions. New exact solutions are constructed under parameter constraints derived from the bifurcation analysis. The influence of physical parameters on these solutions is examined, and graphical representations are provided. Furthermore, the effects of external periodic forcing are explored, revealing quasi‐periodic and chaotic dynamics through phase portraits, Poincaré sections, and Lyapunov exponents. Numerical simulations demonstrate that the system's behavior is highly sensitive to the frequency and amplitude of the external forcing.