  We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility
condition involves the hyperbolic operator, the fundamental phase associated
with the initial oscillation, and the semilinear source term; it states roughly
that hyperbolicity is preserved around resonances.
  If the compatibility condition is satisfied, the solutions are defined over
time intervals independent of the wavelength, and the associated WKB solutions
are stable under a large class of initial perturbations. If the compatibility
condition is not satisfied, resonances are exponentially amplified, and
arbitrarily small initial perturbations can destabilize the WKB solutions in
small time.
  The amplification mechanism is based on the observation that in frequency
space, resonances correspond to points of weak hyperbolicity. At such points,
the behavior of the system depends on the lower order terms through the
compatibility condition.
  The analysis relies, in the unstable case, on a short-time Duhamel
representation formula for solutions of zeroth-order pseudo-differential
equations.
  Our examples include coupled Klein-Gordon systems, and systems describing
Raman and Brillouin instabilities.
