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

Abstract

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.

Reviving Modulational Instability with Third-Order Dispersion

“…It is to noted that we have also studied the dispersion relation for the system with HEM (Λ = 1) and observed that the range of forbidden frequencies is even more when compared to the system with LEM (Λ = 0.01) albeit they exhibit somewhat different dispersion curves (not shown here). These findings clearly suggest that the considered systems may support some novel kinds of localized states (as the dispersion relation of the present system is having analogy with the cryptographic system [20,21]) which we leave for a detailed subsequent work.…”

El juego de los abalorios: del Último Teorema de Fermat a los Solitones Ópticos

“…La propagación de ondas y los modelos no lineales son temas que aparecen en muchísimos sistemas: en el aire [8], en el agua [9], en los sólidos [10] y, de manera particularmente importante, en sistemas no lineales en los cuales puede propagarse la luz [11]. En este trabajo nos interesan los modelos que describen la propagación de pulsos de luz en fibras ópticas.…”

EL JUEGO DE LOS ABALORIOS: del Último Teorema de Fermat a los Solitones Ópticos

“…Si examinamos las 5 ecuaciones mencionadas en la sección anterior encontraremos que en esos 5 casos las relaciones de dispersión están dadas por ecuaciones de la forma: Estas curvas son inesperadas. Nunca antes de la publicación de la referencia [20] se habían reportado relaciones de dispersión con estas formas. La curva que se observa en la Fig.…”

“…La curva que se observa en la Fig. 2 de [20] es particularmente interesante. De acuerdo a esta curva hay 2 intervalos de frecuencia prohibidos:…”

“…En las Figs. 1 y 2 de la Ref [20]. podemos ver las formas que toma esta curva cuando (𝑐 0 , 𝑐 2 , 𝑐 3 ) = (1/30, 1/2, 1/15) y cuando (𝑐 0 , 𝑐 2 , 𝑐 3 ) = (1/40, 1/2, 1/50).…”