Diffractive Nonlinear Geometric Optics with Rectification

This paper studies high frequency solutions of nonlinear hyperbolic equations for time scales at which diiractive eeects and nonlinear eeects are both present in the leading term of approximate solutions. The key innovation is the analysis of rectiication eeects, that is the interaction of the nonoscillatory local mean eld with the rapidly oscillating elds. The main results prove that in the limit of frequency tending to innnity, the relative error in our approximate solutions tends to zero. One of our main conclusions is that for oscillatory elds associated with wave vectors on curved parts of the characteristic variety, the interaction is negligible to leading order. For wave vectors on at parts of the variety, the interaction is spelled out in detail. Outline. x1. Introduction. x2. The ansatz and the rst proole equations. x3. Large time asymptotics for linear symmetric hyperbolic systems. x4. Proole equations, continuation. x5. Solvability of the proole equations. x6. Convergence and stability. x7. Applications, examples and extensions. References x1. Introduction. This paper continues the study, initiated in DJMR], D], of the behavior of high frequency solutions of nonlinear hyperbolic equations for time scales at which diiractive eeects and nonlinear eeects are both present in the leading term of approximate solutions. By diiractive eeects we mean that the leading term in the asymptotic expansion has support which extends beyond the region reached by the rays of geometric optics. Our expansions are for problems where the time scale for nonlinear interaction is comparable to the time scale for the onset of diiractive eeects. The key innovation is the analysis of rectiication eeects, that is the interaction of the nonoscillatory local mean eld with the rapidly oscillating elds. On the long time scales associated with diiraction, these nonoscillatory elds tend to behave very diierently from the oscillating elds. One of our main conclusions is that for oscillatory elds associated with wave vectors on curved parts of the characteristic variety, the interaction is negligible to leading order. For wave vector on at parts of the variety, and in particular problems for problems in one dimensional space, the interaction 1 cannot be ignored and is spelled out in detail. In all cases the leading term in an approximation is rigorously justiied, the correctors are o(1) as the wavelength " tends to zero. The error is not O(") with > 0. The point of departure are the papers DJMR], D] where innnitely accurate …