Solitary waves of the regularized short pulse and Ostrovsky equations

TL;DR

This paper introduces a regularized model for short pulse propagation in nonlinear media, demonstrating the existence of smooth solitary waves similar to modified KdV solitons through advanced mathematical analysis.

Contribution

The paper derives a higher order regularization of the short pulse equation and proves the existence of smooth solitary wave solutions using Fenichel theory and Melnikov analysis.

Findings

Existence of smooth traveling waves in the regularized model

Comparison with solitary waves of the modified KdV equation

Extension of results to 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 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.