Modulated wave train with oblique sidebands in finite water depths

In this study, with the prediction directly from the framework based on the Zakharov equation using kernels in Hamiltonian form, instability regions were obtained in different water depths. Extensive numerical simulations were conducted using a higher-order spectral method to investigate the modulational instability. Numerical simulations confirm that sidebands grow exponentially, suggesting that there is noticeable modulational instability in shallow water at dimensionless water depth 0.8. An evident amplification can also be achieved.