Nonlinear Geometric Optics for Short Pulses

This paper studies the propagation of pulse-like solutions of semilinear hyperbolic equations in the limit of short wavelength. The pulses are located at a wavefront Σ {φ=0} where φ satisfies the eikonal equation and dφ lies on a regular sheet of the characteristic variety. The approximate solutions are u approx=U (t, x, φ(t, x)/ ) where U(t, x, r) is a smooth function with compact support in r. When U satisfies a familiar nonlinear transport equation from geometric optics it is proved that there is a family of exact solutions u exact such that u approx has relative error O( ) as →0. While the transport equation is familiar, the construction of correctors and justification of the approximation are different from the analogous problems concerning the propagation of wave trains with slowly varying envelope.