Ultrashort pulses and short-pulse equations in $(2+1)-$dimensions

TL;DR

This paper derives and analyzes two versions of the (2+1)-dimensional short pulse equation from Maxwell's equations, exploring their properties, conservation laws, and numerical behaviors relevant to advanced optical materials.

Contribution

It introduces two novel (2+1)-dimensional short pulse equations derived from Maxwell's equations, including their mathematical properties and numerical simulations.

Findings

Ultrashort 1D breathers are robust.

2D localized initial conditions disperse into quasi-1D waveforms.

The models exhibit fundamental conservation laws and Hamiltonian structure.

Abstract

In this paper, we derive and study two versions of the short pulse equation (SPE) in $(2+1)-$dimensions. Using Maxwell's equations as a starting point, and suitable Kramers-Kronig formulas for the permittivity and permeability of the medium, which are relevant, e.g., to left-handed metamaterials and dielectric slab waveguides, we employ a multiple scales technique to obtain the relevant models. General properties of the resulting $(2+1)$-dimensional SPEs, including fundamental conservation laws, as well as the Lagrangian and Hamiltonian structure and numerical simulations for one- and two-dimensional initial data, are presented. Ultrashort 1D breathers appear to be fairly robust, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing waveforms.