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

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 wave guides, 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 one-dimensional breathers appear to be fairly robust, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing wave forms.

Modeling of ultrashort pulse propagation in lossy nonlinear metamaterials

“…Finally, it would also be of particular interest to extend our considerations to higher‐dimensional settings (e.g., Reference for a recent work related to the SPE model) and study – apart from solitons – other physically relevant and important phenomena, like collapse, vortex formation and their dissipative dynamics, and so on.…”

The determinant representation of an N-fold Darboux transformation for the short pulse equation

“…[22], in which φ is a solution of the SG equation, and this idea is used to get periodic solutions of the SP equation [23] and the multisoliton solutions of the multi-component SP equation [24]. There exist several extensions of the SP equation including the discrete SP equation [6], the two-component SP [7,34], multi-component SP [24], stochastic SP [18], (2+1)-dimensional SP [31], complex SP [8,21], etc.…”

Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms

“…The SPE was later used in the study of localized structures in frequency band gaps of nonlinear metamaterials [30]. Recently, two different versions of the SPE in (2 + 1) dimensions were studied in detail [31] and it was shown that ultrashort one-dimensional breathers appear to be fairly robust in the two-dimensional setting, while rather general two-dimensional localized initial conditions are transformed into quasi-one-dimensional dispersing wave forms, which are reminiscent of the one-dimensional solitons.…”

“…The nonlinear effects associated with the spatiotemporal propagation of few-cycle optical pulses in nonlinear dispersive media, including nonlinearity-induced self-phase modulation, generation of higher harmonics, and the effects of diffraction, were analyzed in detail [46]. A detailed study of ultrashort pulses and short-pulse equations in (2 + 1) dimensions was also performed in a recent work by Shen et al [31]. Two versions of the short pulse equation in (2 + 1) dimensions were derived and by 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, a multiple scales technique to obtain the relevant models was employed [31].…”

“…A detailed study of ultrashort pulses and short-pulse equations in (2 + 1) dimensions was also performed in a recent work by Shen et al [31]. Two versions of the short pulse equation in (2 + 1) dimensions were derived and by 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, a multiple scales technique to obtain the relevant models was employed [31]. Kolesik et al [47] quantified the limits of unidirectional ultrashort optical pulse propagation, explored the limits of the unidirectional pulse propagation equation in general nonlinear media, and investigated under which physical conditions two-way propagation becomes significant, and leads to a breakdown of the unidirectional approximation.…”