We show that a simple compatibility condition determines the
qualitative behavior of the solutions to semilinear, dispersive,
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 interactions coefficients are not too large at the 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.

Our examples include coupled Klein-Gordon equations in semilinear case.