Chaotic dynamic characteristics during the long time evolution of wave trains

The long time evolution of wave trains is of great significance for understanding the fundamental laws and intrinsic mechanisms of nonlinear water wave dynamics. However, the state of wave field and chaotic dynamic characteristics during the long time evolution process remain unrevealed. Employing the High-Order Spectral method, combining with chaos theory, these characteristics are investigated both qualitatively and quantitatively. The results reveal that wave field transitions from a stable to a chaotic state over time. In the stage of modulation instability, phase space trajectories cluster within a closed region, indicating a stable and predictable wave field. However, after long time evolution, trajectories become disordered, featuring strange attractors, a positive Largest Lyapunov Exponent (LLE), a non-integer Correlation Dimension (CD), and positive, finite Kolmogorov Entropy (KS), all of which signify a chaotic wave field. The initial wave steepness ε0 and the length of evolution time both significantly influence the wave field state in the evolution. Within a specific range of ε0, CD, LLE, and KS all positively correlate with ε0. As ε0 increases, the nonlinearity intensifies, and the long time evolution of wave trains becomes more sensitive to initial conditions, resulting in more significant chaotic behavior and higher levels of chaos.