Families of Bragg-grating solitons in a cubic–quintic medium

Atai, Javid; Malomed, Boris A.

doi:10.1016/s0375-9601(01)00314-0

Cited by 114 publications

(44 citation statements)

References 21 publications

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“…These simulations at κ = 1/2 reinforce our belief that our two dynamical criteria for understanding the stability of the solitary wave, namely the stability curve p(v) as well as studying orbits in the phase portrait are accurate indications of the stability of the solitary waves as obtained by numerical simulations of the FNLSE. Our results are likely to shed light on the behavior of a number of physical systems varying from optical fibers [1], Bragg gratings [2], BECs [3,4], nonlinear optics, and photonic crystals [7].…”

Section: Discussionmentioning

confidence: 94%

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Forced nonlinear Schrödinger equation with arbitrary nonlinearity

et al. 2012

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We consider the nonlinear Schrödinger equation (NLSE) in 1 + 1 dimension with scalar-scalar self-interaction g 2 κ+1 (ψ ψ) κ+1 in the presence of the external forcing terms of the form re −i(kx+θ) − δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v k = 2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r → 0. In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that dp(t)/dq(t) < 0, where p(t) is the normalized canonical momentum p(t) = 1 M(t)∂L ∂q , anḋ q(t) is the solitary wave velocity. Here M(t) = dxψ (x,t)ψ(x,t). Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.

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Section: Discussionmentioning

confidence: 94%

“…The forced nonlinear Schrödinger equation (FNLSE) for an interaction of the form (ψ ψ) 2 has been recently studied [8,9] using collective coordinate (CC) methods such as timedependent variational methods and the generalized traveling wave method (GTWM) [10]. In Refs.…”

Section: Introductionmentioning

confidence: 99%

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

et al. 2012

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“…Equation (1) is scaled so as to make the effective cubic coefficient equal to 1, while is respective quintic coefficient, proportional to / . As we choose the self-focusing cubic and defocusing quintic nonlinearities, which provides for the stabilization of solitons [50,51], is positive. If the pulse's temporal width is larger than the intra-band relaxation time, the gain spectrum, ( ), can be expanded in the Taylor series about the carrier frequency .…”

Section: The Modelmentioning

confidence: 99%

Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion

2017

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We study the generation of dissipative solitons (DSs) in the model of the fiber-laser cavities under the combined action of cubic-quintic nonlinearity, multiphoton absorption and/or multiphoton emission (nonlinear gain) and gain dispersion. A random component of the group-velocity dispersion (GVD) is included too. The DS creation and propagation is studied by means of a variational approximation and direct simulations, which are found to be in reasonable agreement. With a proper choice of the gain, robust DS operation regimes are predicted for different combinations of multiphoton absorption and emission, in spite of the presence of the perturbation in the form of the random GVD. Importantly, the zero background around the solitons remains stable in the presence of the (necessary) linear gain. The solitons are stable too against a certain (realistic) level of noise. Another essential finding is that the quintic gain in the form of three-photon emission (3PE) offers an alternative mechanism for supporting stable solitons, provided that it is not too strong. The DSs coexist in low-and highamplitude forms, for a given value of their width. The low-amplitude DS is stable, while its high-amplitude counterpart is subject to the blowup instability, in the presence of the 3PE. Interactions between DSs show various scenarios of the creation of breather states through merger of the two solitons.

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“…(3)- (6), generates cubic SPM and XPM terms in the lowest approximation, and various quintic terms as first corrections to it. In this connection, it is relevant to mention that gap solitons in the coupled-mode BG equations, including self-focusing cubic and selfdefocusing quintic terms, were studied in [38]. Two different types of gap solitons were identified in that work, namely, regular and "two-tiered" ones, dominated by the cubic and quintic nonlinearity, respectively.…”

Section: ͑7͒mentioning

confidence: 99%

“…while the amplitude of Im͓u͑x͔͒ remains finite (the saturable nonlinearity, unlike the above-mentioned cubicquintic one [38], admits solitons with an arbitrarily large amplitude). Simultaneously, the soliton is much broader than its counterpart in the standard model.…”

Section: Families Of Bragg-grating Solitons In the Model With Saturabmentioning

confidence: 99%

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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Families of Bragg-grating solitons in a cubic–quintic medium

Cited by 114 publications

References 21 publications

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

Forced nonlinear Schrödinger equation with arbitrary nonlinearity

Dissipative soliton fiber lasers with higher-order nonlinearity, multiphoton absorption and emission, and random dispersion

Solitons in Bragg gratings with saturable nonlinearities

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