Solitary Waves of the Regularized Short Pulse and Ostrovsky Equations

Abstract

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 Korteweg–deVries 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.