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- W2963199176 abstract "— 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. 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." @default.
- W2963199176 created "2019-07-30" @default.
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- W2963199176 date "2015-01-01" @default.
- W2963199176 modified "2023-09-30" @default.
- W2963199176 title "A Stability Criterion for High-Frequency Oscillations" @default.
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