Wave turbulence: the case of capillary waves

Capillary waves are perhaps the simplest example to consider for an introduction to wave turbulence. Since the first paper by Zakharov and Filonenko, capillary wave turbulence has been the subject of many studies, but a didactic derivation of the kinetic equation is still lacking. It is the objective of this paper to present such a derivation in the absence of gravity and in the approximation of deep water. We use the Eulerian method and a Taylor expansion around the equilibrium elevation for the velocity potential to derive the kinetic equation. The use of directional polarities for three-wave interactions leads to a compact form for this equation which is fully compatible with previous work. The exact solutions are derived with the so-called Zakharov transformation applied to wavenumbers, and the nature of these solutions is discussed. Experimental and numerical works done in recent decades are also reviewed.