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Appl. Sci. 2017, 7(4), 340; doi:10.3390/app7040340

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

1
Instituto de Física, Dpto. de Sistemas Complejos, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México D.F., Mexico
2
Instituto de Física, Dpto. de Materia Condensada, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México D.F., Mexico
3
Facultad de Ciencias, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México D.F., Mexico
*
Author to whom correspondence should be addressed.
Academic Editor: Boris Malomed
Received: 23 December 2016 / Revised: 25 March 2017 / Accepted: 26 March 2017 / Published: 30 March 2017
(This article belongs to the Special Issue Guided-Wave Optics)

Abstract

We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation direction) which appears when we abandon the slowly varying envelope approximation. We show that as a consequence of these forbidden bands, narrow pulses radiate in a novel and peculiar way. We also show that the nonlinear equations studied in this paper have exact soliton-like solutions of different forms, some of them being embedded solitons. The solutions obtained (of the linear as well as the nonlinear equations) are interesting since several arguments suggest that the Cauchy problems for these equations are ill-posed, and therefore the specification of the initial conditions is a delicate issue. It is also shown that some of these equations are related to elliptic curves, thus suggesting that these equations might be related to other fields where these curves appear, such as the theory of modular forms and Weierstrass ℘ functions, or the design of cryptographic protocols. View Full-Text
Keywords: optical solitons; embedded solitons; soliton radiation; nonlinear Schrödinger equation; elliptic curves; forbidden frequencies; spectral gap optical solitons; embedded solitons; soliton radiation; nonlinear Schrödinger equation; elliptic curves; forbidden frequencies; spectral gap
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Fujioka, J.; Gómez-Rodríguez, A.; Espinosa-Cerón, Á. Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion. Appl. Sci. 2017, 7, 340.

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