Energy spectrum of ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

In this Letter we regard nonlinear gravity-capillary waves with parameter of nonlinearity being $\varepsilon \sim 0.1 \div 0.25$. For this nonlinearity time scale separation does not occur and kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in \emph{Kartashova, \emph{EPL} \textbf{97} (2012), 30004.} We compute for the first time an analytical expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function depending on the ratio of surface tension to the gravity acceleration. It is shown that its two limits - pure capillary and pure gravity waves on a fluid surface - coincide with the previously obtained results. We also discuss relations of the model of D-cascade with a few known models used in the theory of nonlinear waves such as Zakharov's equation, resonance of the modes with nonlinear Stokes corrected frequencies and Benjamin-Feir index. These connections are crucial in the understanding and forecasting specifics of the energy transport in a variety of multi-component wave dynamics, from oceanography to optics, from plasma physics to acoustics.