Periodic multiphase nonlinear diffractive optics with curved phases

Abstract

We describe diffraction for rapidly oscillating, periodically modulated nonlinear waves. This phenomenon arises for example when considering long-time propagation, or through perturbation of initial oscillations. We show existence and stability of solutions to variable coefficient, nonlinear hyperbolic systems, together with 3-scale multiphase infinite-order WKB asymptotics: the fast scale is that of oscillations, the slow one describes the modulation of the envelope, which is along rays for the oscillatory components, and the intermediate one corresponds to transverse diffraction. It gives rise to nonlinear Schrodinger equations on a torus for the profiles. The main difficulty resides in the fact that the coefficients in the original equations are variable: thus, phases are nonlinear, and rays are not parallel lines. This induces variable coefficients in the integro-differential system of profile equations, which in general is not solvable. We give sufficient (and, in general, necessary) geometrical coherence conditions on the phases for the formal asymptotics to be rigorously justified. Small divisors assumptions are also needed, which are generically satisfied.