Transformation and integrability of a generalized short pulse equation

TL;DR

This paper demonstrates the integrability of a generalized short pulse equation in two specific cases, providing new insights into its structure and solutions, including the shortest possible envelope soliton.

Contribution

It identifies two integrable cases of the generalized short pulse equation and derives their Lax pairs and bi-Hamiltonian structures, highlighting the overlooked single-cycle pulse equation.

Findings

The generalized short pulse equation is integrable in two cases.

The single-cycle pulse equation (SCPE) is a new integrable reduction.

The SCPE admits envelope solitons as short as one cycle.

Abstract

By means of transformations to nonlinear Klein-Gordon equations, we show that a generalized short pulse equation is integrable in two (and, most probably, only two) distinct cases of its coefficients. The first case is the original short pulse equation (SPE). The second case, which we call the single-cycle pulse equation (SCPE), is a previously overlooked scalar reduction of a known integrable system of coupled SPEs. We get the Lax pair and bi-Hamiltonian structure for the SCPE and show that the smooth envelope soliton of the SCPE can be as short as only one cycle of its carrier frequency.