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WKB asymptotics for the Euler-Maxwell equations. (English) Zbl 1116.35114
Summary: The Euler-Maxwell system of equations is a complex, hydrodynamical model for the description of laser-plasma interactions. We introduce non-dimensional variables, exhibit a small parameter and study various WKB approximations for this system, under a polarization condition for the initial data. We justify an approximation by a weak Zakharov equation for times $$O(1)$$ and an approximation by a Davey-Stewartson equation for times $$O(|\log|)$$. Our key observation is that the Euler-Maxwell system has transparency properties, similar to the properties exhibited by Joly, Métivier and Rauch for Maxwell-Bloch systems. These properties imply in particular that in a weakly nonlinear regime, the geometric optics approximation is given by a linear equation.

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 78A60 Lasers, masers, optical bistability, nonlinear optics 35C20 Asymptotic expansions of solutions to PDEs 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory