Numerical simulation of surface waves instability on a homogeneous grid

Highlights: • We described three and four wave instabilities due to resonant interactions. • Algorithm for simulation of weakly nonlinear surface waves is presented. • Numerical scheme conserves Hamiltonian of the system. • We discussed and simulated instability of standing and propagating waves. We performed full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence. Instability of both propagating and standing waves was studied. We separately studied pure capillary wave, that was unstable due to three-wave interactions and pure gravity waves, that were unstable due to four-wave interactions. The theoretical description of instabilities in all cases is included in the article. The numerical algorithm used in these and many other previous simulations performed by the authors is described in detail.