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 Schrödinger equation, which models slowly varying wavetrains, this equation has no smooth pulse solutions which propagate with fixed shape and speed.