Space-time resonances and high-frequency Raman instabilities in the two-fluid Euler-Maxwell system

We show that space-time resonances induce high-frequency Raman instabilities in the two-fluid Euler-Maxwell system describing laser-plasma interactions. A consequence is that the Zakharov WKB approximation to Euler-Maxwell is unstable for non-zero group velocities. A key step in the proof is the reformulation of the set of resonant frequencies as the locus of weak hyperbolicity for linearized equations around the WKB solution. We analyze those linearized equations with the symbolic flow method. Due to large transverse variations in the WKB profile, the equation satisfied by the symbolic flow around resonant frequencies is a linear {\it partial} differential equation. At space-time resonances corresponding to Raman instabilities, we observe a fast growth of the symbolic flow, which translates into an instability result for the original system. The strongest instability is caused by backscattered Raman waves.