Weak turbulence theory for reactive instability

In the present paper, the customary weak turbulence theory is generalized to include reactive instabilities. For the sake of simplicity, the formalism assumes electrostatic perturbation propagating in one-dimensional uniform unmagnetized plasmas. By weak turbulence theory it is meant as the perturbative nonlinear theory based upon Vlasov equation, truncated at the second (or up to third) order nonlinearity and ensemble averaged. By reactive instability it is meant as the plasma instability whose growth rate is not necessarily exceedingly small. The traditional weak turbulence theory found in the literature is applicable only to weakly growing plasma instabilities whose real frequency ωk can be determined from the real part of the dispersion relation, Re ε(k,ωk)=0, while the growth rate may be determined by the Landau formula, γk=-Im ε(k,ωk)[∂ Re ε(k,ωk)/∂ωk]-1. This implies the assumption that |γk|<<ωk. On the other hand, for reactive instabilities for which γk/ωk is not necessarily small, the real frequency and growth/damping rate must be determined from the complex roots of the dispersion relation, ε(k,ωk+iγk)=0. The present paper extends the textbook weak turbulence theory to deal with such a situation.