Turbulent 4-wave Interaction of Two Type of Waves

We consider turbulent 4-wave interaction of two types of waves: acoustic waves (dispersion $\omega = k$) and electromagnetic-type waves (dispersion $\Omega^2 = m^2 + p^2$). For large wave vectors ($k \gg m$), when the dispersion of EM-type waves becomes quasiacoustic, the stationary spectra are obtained following a standard Zakharov approach. In the small wave number region ($k, p \ll m$) we derive nonlinear differential Kompaneets-type kinetic equation for arbitrary distributions of the interacting fields and find when this equation has Kolmogorov-type power law solutions.