On the Boyd-Kadomstev System for a Three-Wave Coupling Problem and its Asymptotic Limit

We consider the Boyd-Kadomstev system which is in particular a model for the Brillouin backscattering in laser-plasma interaction. It couples the propagation of two laser beams, the incoming and the backscattered waves, with an ion acoustic wave which propagates at a much slower speed. The ratio \({\varepsilon}\) between the plasma sound velocity and the (group) velocity of light is small, with typical value of order 10−3. In this paper, we make a rigorous analysis of the behavior of solutions as \({\varepsilon \to 0}\) . This problem can be cast in the general framework of fast singular limits for hyperbolic systems. The main new point which is addressed in our analysis is that the singular relaxation term present in the equation is a nonlinear first order system.