A unified qualitative and data-driven framework for chaotic dynamics in the nonlinear Benney-Luke equation

This study presents a unified qualitative and data-driven framework for investigating chaotic dynamics in the nonlinear Benney-Luke equation, a fundamental model of dispersive long-wave propagation. A travelling-wave transformation reduces the governing partial differential equation to an autonomous dynamical system. The resulting planar reduction is shown to possess a conservative Hamiltonian structure with two equilibrium points, namely a centre and a saddle, whose classifications are determined by the sign of the reduced coefficient \\(\\epsilon _{2}\\) . Since the planar system cannot support deterministic chaos, the original fourth-order travelling-wave equation is subsequently reformulated as a four-dimensional autonomous system and investigated using phase-space trajectories, time-series analysis, largest Lyapunov exponents, bifurcation diagrams, sensitivity to initial conditions, and Poincaré sections. For the representative parameter set \\(\\alpha =0.5\\) , \\(\\beta =0.3\\) , and \\(c=5.98\\) , the oscillatory states \\(x_{2}\\) , \\(x_{3}\\) , and \\(x_{4}\\) exhibit bounded, irregular, and aperiodic behavior, together with positive largest Lyapunov exponents, complex bifurcation patterns, rapid trajectory divergence, and scattered Poincaré-section points, collectively confirming deterministic chaos. The sign change of \\(\\alpha -\\beta c^{2}\\) occurs near \\(\\beta \\approx 0.014\\) and determines the emergence of a locally unstable direction, whereas a pronounced increase in chaotic intensity is observed near \\(c\\approx 6.28\\) . Furthermore, a Nonlinear Autoregressive Neural Network with Exogenous Input is trained using the Levenberg-Marquardt algorithm on 2000 sequential samples, divided into 70%, 15%, and 15% subsets for training, validation, and testing, respectively. The trained network accurately reconstructs the temporal evolution and phase-space geometry of the system, achieving a best validation mean squared error of \\(4.176\\times 10^{-5}\\) and a regression coefficient close to unity, \\(R\\approx 1\\) , with prediction errors concentrated near zero. These findings demonstrate that the proposed analytical-computational framework can effectively characterize and predict chaotic dynamics in the Benney-Luke equation and may be extended to other nonlinear dispersive wave models.