Dynamics of solitons and quasisolitons of the cubic third-order nonlinear Schrödinger equation

We develop a theory of the soliton self-frequency shift compensation by the resonant radiation recently observed in photonic crystal fibers. Our approach is based on the calculation of the soliton plus radiation solution of the generalized nonlinear Schrödinger (GNLS) equation and on subsequent use of the adiabatic theory leading to a system of equations governing evolution of the soliton parameters in the presence of the Raman effect and radiation. Our theoretical results are found to be in good agreement with direct numerical modeling of the GNLS equation.

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“…Other differences between the two types of quasisolitons have also been recently discussed in Ref. [42].…”

Section: Discussionmentioning

confidence: 99%

Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers

2004

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“…(1) was pointed out by Ablowitz and Segur as early as 1981 [29]. The precise form of the bright solitons in the particular case when ε = 6γ was presented recently by Karpman et al [30].…”

Section: Double Embedded Solitonsmentioning

confidence: 97%

“…[30] it was pointed out that Eq. (1) is a particular case of a more general NLS-like equation possessing ESs.…”

Section: Double Embedded Solitonsmentioning

confidence: 99%

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Standard and embedded solitons in nematic optical fibers

et al. 2003

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A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wavepackets of transverse magnetic (T M ) modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations (P DEs) which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive a complex modified Korteweg-de Vries equation (cmKdV) which governs the dynamics for the amplitude of the wavepacket. In this derivation the dispersion, self-focussing and diffraction in the nematic are taken * Fellow of SNI, Mexico † Correspondence author. E-mail: zepeda@fisica.unam.mx ‡ Fellow of SNI, Mexico 1 into account. It is shown that this cmKdV equation has two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double embedded solitons. We explain why these solitons do not radiate at all, even though their wavenumbers are contained in the linear spectrum of the system. We study (numerically and analytically) the stability of these solitons. Our results show that these embedded solitons are stable solutions, which is an interesting property since in most systems the embedded solitons are weakly unstable solutions. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.

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“…(a) −iu ttt and/or u 4t : these higher-order derivatives are necessary to describe sub-picosecond pulses [1][2][3][4][5][6][7][8][9][10][11][12]; in particular, conditions for including u 4t and discarding −iu ttt are discussed in [7,9], (b) |u| 4 u: this higher-order nonlinearity is used when we want to describe the propagation of pulses when the light intensity approaches the values which produce the "saturation" of the refractive index [13][14][15][16][17][18][19][20], (c) i(|u| 2 u) t : this term is necessary to describe the self-steepening of the optical pulses [21][22][23], (d) u(|u| 2 ) t : this term is associated with the effect of Raman scattering [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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

2017

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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.

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Dynamics of solitons and quasisolitons of the cubic third-order nonlinear Schrödinger equation

Cited by 50 publications

References 28 publications

Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers

Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers

Standard and embedded solitons in nematic optical fibers

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

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