On the stability of Kolmogorov spectra of a weak turbulence

The stability problem of Kolmogorov spectra of a weak turbulence is analytically solved for the first time in the framework of a three-wave kinetic equation. The spectrum of isotropic perturbations of a stationary not-in-equilibrium distribution is found for the capillary waves on a shallow water surface. It is shown, in the isotropic case, that the Kolmogorov solution is stable with respect to excitations local in k-space. The perturbations drift to the damping region without growth of the magnitude. The structural instability of the isotropic spectrum is found by computer simulation: a small pumping anisotropy causes the spectrum to be essentially anisotropic within the inertial range.