Propagation of ultra-short optical pulses in cubic nonlinear media

We derive a partial differential equation that approximates solutions of Maxwell's equations describing the propagation of ultra-short optical pulses in nonlinear media,and which extends the prior analysis of Alterman and Rauch [Phys. Lett. A 264 (2000) 390; Diffractive nonlinear geometric optics for short pulses, Preprint, 2002]. We discuss (non-rigorously) conditions under which this approximation should be valid, but the main contributions of this paper are: (1) an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration, (2) a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation, (3) a local well-posedness result for the model equation and (4) a proof that in contrast to the nonlinear Schrodinger equation, which models slowly varying wavetrains, this equation has no smooth pulse solutions which propagate with fixed shape and, speed. (C) 2004 Elsevier B.V. All rights reserved.