Dispersive effects for nonlinear geometrical optics with rectification

Abstract

Oscillating approximate solutions to nonlinear hyperbolic dispersive systems are studied. Ansatz of three scales are used in order to deal with diffractive effects. The scaling of the approximate solutions is chosen so that diffractive, dispersive effects and rectification are present in the leading term.

The propagation along the rays of geometrical optics of the oscillating Fourier coefficients of the leading terms is corrected by a Schrödinger dispersion which appears for long times only. The propagation of the nonoscillating Fourier coefficient depends on the properties of a symmetric hyperbolic system, whose characteristic variety is the tangent cone at $0$ to the characteristic variety of the initial operator.

Equations determining the leading term require a sublinear growth condition for the corrector and the inroduction of the analytical “average operators” which convey this sublinear growth condition in a simple way and sort the nonlinearities out.

In the last part, detailed physical examples are given.