Nonlinear wave group interaction in the long time evolution of wave trains

The long time evolution of wave trains involves various nonlinear stages, with significant differences in the wave group shape at each stage. To further investigate the characteristics of nonlinear wave group interaction during the long time evolution of wave trains, the High-Order Spectral method and wavelet transform analysis are employed, and a novel spatial wave group identification method suitable for long time evolution process is introduced. Then the wave groups in the evolution process are classified into four types based on the wave group length. The results show that during the stage of modulation instability, all wave groups are of Type I, which is a result of modulation instability. In this stage, all wave groups propagate at the same velocity without any energy exchange between them, maintaining independent evolution. The appearance of the other three types of wave groups indicates the presence of nonlinear wave group interaction. Under the dominance of nonlinear wave group interaction, the number and length of wave groups no longer remain constant, with significant changes observed in their characteristic parameters. Additionally, the propagation velocities of the wave groups evolve continuously. When two wave groups with different velocities merge, the resulting group accelerates rather than decelerates. In the subsequent evolution, the participating wave groups begin to separate again, with the wave group that was initially trailing overtaking the one that was leading, and their velocities eventually approaching. It is worth noting that the different types of wave groups are the result of nonlinear interactions and also serve as the fundamental units for the subsequent nonlinear interaction processes.