Conclusion

It is found that the high-order nonlinear interactions play very important roles in the long time evolutions of gravity waves both deep and shallow-water solving the hydrodynamic equations for gravity waves having narrow to broad banded spectra. It is concluded as follows:

• The high-order nonlinear interactions can transfer energy among the Fourier modes and excite apparently chaotic mode evolutions even if wave have broad band spectrum in deep-water.
• The high-order nonlinear interactions can generate a single extreme high wave having outstanding crest height such as a freak wave.
• The high-order nonlinear interactions affect $H_{max}$, kurtosis and $GF$ remarkably.
• The high-order nonlinear interactions increase the occurrence probability of large wave height in deep-water and decrease it in shallow-water in comparison with the Rayleigh theory.

Consequently, the high-order nonlinear effects should be taken into account independently of the spectral bandwidth to predict the maximum wave and the freak wave generation.