On the Kolmogorov–Zakharov spectra of weak turbulence

Zakharov discovered that wave kinetic equations for weak turbulence have exact power-law solutions which are similar to the Kolmogorov spectrum of hydrodynamic turbulence. We present a considerably simplified derivation of these solutions, i.e. Kolmogorov–Zakharov spectra. Even to a greater extent we simplify the derivation of certain “universal” corrections to these spectra, obtained by Kats and Kontorovich. The technique is utilized as well to derive the expressions for the so-called Mellin functions (that describe the behavior of weak turbulence in the vicinity of the Kolmogorov–Zakharov spectra). Using this general approach, we obtain Kolmogorov–Zakharov spectra for a class of weak turbulent media with dissipative nonlinearity. We also find a large family of anisotropic spectra, which are exact solutions of the corresponding kinetic equation. As a physical example, we consider the plasma turbulence when the main nonlinear process is the scattering of plasmons by electrons.