Derivation of the Zakharov Equations

This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MéTIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.