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.Comment: 14 pages, 1 figure, accepted for publication in Physics Letters

Scattering for the Zakharov system in 3 dimensions

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. The wave component is shown to decay pointwise at the optimal rate of t^{-1}, whereas the Schr\"odinger component decays almost at a rate of t^{-7/6}.Comment: Minor changes and referee's comments include