Serie : Mémoires de la Société mathématique de France. Vol 142
Paru le 24/09/2015 | Broché VI-130 pages
Professionnels
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 résonances.
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.
In the unstable case, the key observation is that resonances correspond to weakly hyperbolic frequencies ; the amplification proof then relies 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.
