Propagation of ultra-short optical pulses in 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 [1], [2]. 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 slowing varying wavetrains this equation has no pulse solutions which propagate with fixed shape and speed.

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