Euler’s equation describing the potential flow of an ideal incompressible fluid with a free surface can be formulated in a Hamiltonian form. Expansion of the Hamiltonian in powers of nonlinearity up to the terms of fourth order leads to equations that can be solved numerically by the spectral code. We performed the simulation of these equations on the grids of 256×256 and 512×512 harmonics. To stabilize the algorithm and avoid the accumulation of energy in high wave numbers, we introduced an artificial high-frequency dumping to this equation. The cases of gravity and capillary waves we studied separately. In both cases we studied the development of instability for stationary waves as well as the formation of weak-turbulent Kolmogorov spectra. The stationary capillary waves undergo the first order decay instability. This instability leads to appearing of secondary waves concentrated in K-plane near the curve describing by the resonant condition

ω(~k0) = ω(~k) + ω(~k0 − ~k)

Here ω(k) = |σk| 3/2 and ~k0 is the initial wave vector. The secondary waves, in their turn, undergo the decay instability. The process repeats itself several times and leads to the formation of totally chaotic turbulent state. Stationary gravity waves are unstable with respect to secondary-order (modulation) instability that leads to the formation of waves on the Phillips curve

2ω(~k0) = ω(~k) + ω(2~k0 − ~k)

We have observed this process in our experiments. As the theory of weak turbulence predicts, we observed the formation of weak-turbulent Kolmogorov spectra for both capillary and gravity waves.