Derivation of the Zakharov equations

This paper continues the study, initiated in [28, 8], of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for well-prepared initial data. We apply this result to the Euler-Maxwell equations describing 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 which leading terms are solutions of the Zakharov equations. Because of transparency properties of the EulerMaxwell equations put in evidence in [28], this study is led 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 [13, 14], recent results of Lannes on norms of pseudodifferential operators [15], and a semiclassical, paradifferential calculus. ∗Indiana University, Bloomington, IN 47405; [email protected]. This research was partially supported under NSF grant number DMS-0300487. The author warmly thanks Christophe Cheverry, Thierry Colin, David Lannes, Guy Métivier, and Kevin Zumbrun, for the interest they showed for this work and many very interesting discussions.