  We derive a model for the propagation of short pulses in nonlinear media. The
model is a higher order regularization of the short pulse equation (SPE). The
regularization term arises as the next term in the expansion of the
susceptibility in derivation of the SPE. Without the regularization term there
do not exist traveling pulses in the class of piecewise smooth functions with
one discontinuity. However, when the regularization term is added we show, for
a particular parameter regime, that the equation supports smooth traveling
waves which have structure similar to solitary waves of the modified KdV
equation. The existence of such traveling pulses is proved via the Fenichel
theory for singularly perturbed systems and a Melnikov type transversality
calculation. Corresponding statements for the Ostrovsky equations are also
included.
