Study of a mathematical model for stimulated Raman scattering T. BOUCHERES, T. COLIN, B. NKONGA, B. TEXIER 1 Math´ ematiques appliqu´ es de Bordeaux, Universit´ e Bordeaux 1 and CNRS UMR 5466, 351 cours de la Lib´ eration, 33405 Talence cedex, France A. BOURGEADE 2 CEA-CESTA, BP2, 33114 Le Barp, France, LRC CEA MO3 Abstract We study a semiclassical modelization of the interaction of a laser with a mono-atomic gas. The Maxwell equations are coupled with a three-level version of the Bloch equations. Taking into account the specificities of the laser pulse and of the gas, we introduce small parameters and a dimen- sionless form of the equations. To describe stimulated Raman scattering, we perform a three-scale WKB expansion in the weakly nonlinear regime of geometric optics. The limit system is of Schr¨ odinger-Bloch type. We prove a global existence result for this system and the convergence of its solution toward the solution of the initial Maxwell-Bloch equations, as the parameter of the WKB expansion goes to 0. We put in evidence Raman instability in the one-dimensional case, both theoretically and numerically. 1 Physical context 1.1 Description of the model We consider the interaction of a laser pulse with a mono-atomic gas. As usual, the laser is described by the Maxwell equations[4] , [15]. The matter is described by the quantic Schr¨ odinger equation. The link between these equations is made by the introduction of the polarization. The system is    ∂ t  B + rot  E =0, ∂ t  E − c 2 rot  B = −µ 0 c 2 ∂ t  P, i∂ t Ψ = H Ψ, where  E,  B,  P designate the electric field, the magnetic field, and the (macro- scopic) polarization. µ 0 and c denote respectively the permeability of free space and the velocity of light,  is the Planck constant. Ψ is the wave-function of the matter and H its Hamiltonian. We suppose that the laser pulse propagates in a dilute atomic vapor. Hence, as a first approximation, one can neglect the coupling between the different atoms and take Ψ as the wave-function of one 1 bouchere@math.u-bordeaux.fr, colin@math.u-bordeaux.fr, nkonga@math.u-bordeaux.fr, btexier@math.u-bordeaux.fr 2 bourgeade@cea.fr 1