Helmholtz algebraic solitons

Christian, J. M.; McDonald, GS; Chamorro‐Posada, Pedro

doi:10.1088/1751-8113/43/8/085212

Cited by 14 publications

(14 citation statements)

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“…As this growth is completely unphysical (as it would imply a fictitious energy growth), the coefficient a(ω) has to be equal to zero, and consequently from Equation (36) it follows that there is an unexpected relationship between U z (0, ω) and U(0, ω):…”

Section: Solutions Of Equationmentioning

confidence: 99%

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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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Section: Solutions Of Equationmentioning

confidence: 99%

“…In particular, the following non-paraxial extensions of the spatial NLS equation have been studied [34][35][36]:…”

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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“…3) are distinct from their competitive-focusing counterparts. 23,24 Previously, the VK criterion has been successfully deployed in the analysis of Helmholtz solitons. 23,24,38,39 The validity of this approach lies in spatial symmetry (see Fig.…”

Section: Stability Criterionmentioning

confidence: 99%

“…We classify these solitons as stable limitcycle attractors of the system. 23,24,38,39 We now analyse solitons in the conditionally-stable regime (where 2 < σ < 4) in terms of the power of the input beam, denoted by P in . Typical curves in the (β, P) plane for solution (20) are shown in Fig.…”

Section: Numerical Perturbation Analysismentioning

confidence: 99%

“…The case n σ > 0, n 2σ < 0 supports bright solitons and a class of nonlinear boundary wave; 23 when n σ < 0, n 2σ > 0, one finds coexisting bright hyperbolic solitons, algebraic (bright and dark) solitons, and also class of nonlinear periodic wave. 24 The stability properties of those various solutions have been characterized by semi-analytical and computational investigations. For completeness, we now explore the remaining scenario that can be expected to support bright solitons: the purely focusing nonlinearity, where n σ > 0 and n 2σ > 0.…”

Section: Introductionmentioning

confidence: 99%

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Helmholtz Solitons in Optical Materials With a Dual Power-Law Refractive Index

Christian

McDonald

Chamorro‐Posada³

2010

J. Nonlinear Optic. Phys. Mat.

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A nonlinear Helmholtz equation is proposed for modelling scalar optical beams in uniform planar waveguides whose nonlinear refractive index exhibits a purely-focusing dual powerlaw dependence on the electric field amplitude. Two families of exact analytical solitons, describing forward-and backward-propagating beams, are derived. These solutions are physically and mathematically distinct from those recently discovered for related nonlinearities. The geometry of the new solitons is examined, conservation laws are reported, and classic paraxial predictions are recovered in a simultaneous multiple limit. Conventional semi-analytical techniques assist in studying the stability of these nonparaxial solitons, whose propagation properties are investigated through extensive simulations.

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