Gap Solitons

Sterke, C. Martijn de; Sipe, J. E.

doi:10.1016/s0079-6638(08)70515-8

Cited by 283 publications

(315 citation statements)

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“…The latter characteristic may be defined as the inverse reflection length; usually, it is 1 / l refl ͑Bragg͒ ϳ 1 mm −1 , for weak gratings [1]. Since the propagation length necessary for the formation and detection of gap solitons amounts to several l refl ͑Bragg͒ [3], the typical longitudinal size of available waveguides, several centimeters, will be quite sufficient for experiments with the spatial BG solitons in this setting. Note that Eqs.…”

Section: ͑7͒mentioning

confidence: 99%

“…An especially interesting manifestation of the nonlinearity is the formation of BG solitons (alias gap solitons, if they are spectrally centered inside the bandgap). A standard theoretical model of a Kerrnonlinear medium equipped with the BG is based on a system of coupled-mode equations (CMEs) for amplitudes of the counterpropagating waves, U͑x , t͒ and V͑x , t͒, which are coupled linearly by the BG reflection, and nonlinearly by the cross-phase modulation (XPM), and also take into account the self-phase modulation (SPM) effect [3]: iU t + iU x + ͑ ͉U͉ 2 + ͉V͉ 2 ͒U + V = 0,…”

Section: Introduction and The Modelsmentioning

confidence: 99%

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Solitons in Bragg gratings with saturable nonlinearities

et al. 2007

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We introduce two different systems of coupled-mode equations to describe the interaction of two waves coupled by the Bragg reflection in the presence of saturable nonlinearity. The basic model assumes the ordinary linear coupling between the modes. It may be realized as a photorefractive waveguide, with a Bragg lattice permanently written in its cladding. We demonstrate the presence of a cutoff point in the system's bandgap, with gap solitons existing only on one side of it. Close to this point, the soliton's norm diverges with power −3 / 2. The soliton family between the cutoff point and the edge of the bandgap is stable. In this model, stationary bound states of two in-phase solitons are found too, but they are unstable, transforming themselves into breathers. Another model assumes a photoinduced longitudinal bulk grating, with the corresponding intermode coupling subject to saturation along with the nonlinearity. In that model, another cutoff point is found, with the soliton's norm diverging near it with power −2. Solitons are stable in this model too (while it does not give rise to twosoliton bound states). Collisions between moving solitons are always quasi-elastic, in either model.

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Section: ͑7͒mentioning

confidence: 99%

Section: Introduction and The Modelsmentioning

confidence: 99%

Solitons in Bragg gratings with saturable nonlinearities

et al. 2007

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“…In situations governed by higher-order or multi-component partial differential equation (PDE) models in 1+1 dimensions, the spectrum of linearised waves generally has at least two branches. If these branches do not fill out the entire possible spectrum of wave frequencies, then one has the possibility of a gap in the linear spectrum where exponentially localised solutions can exist, so called gap solitons [12]. Such solutions can be linearly stable solutions of the PDE in that they are attractors for a range of initial data.…”

Section: Overviewmentioning

confidence: 99%

When gap solitons become embedded solitons: a generic unfolding

Wagenknecht

Champneys

2003

Physica D: Nonlinear Phenomena

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A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z 2 -symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of 5th-order Korteweg de Vries equation.

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“…Detailed theoretical studies of these Bragg-grating (Kerr-nonlinear) solitons (see de Sterke and Sipe [1994] for a review) were followed by their experimental observation (Eggleton, Slusher, de Sterke, Krug and Sipe [1996]) in a short (< 10 cm) piece of an optical fiber with a resonant Bragg grating written on it. In theoretical considerations of solitons in a Bragg grating combined with Kerr nonlinearity, a formidable problem is their stability.…”

Section: Introduction § § § Introductionmentioning

confidence: 99%