Dissipative ring solitons with vorticity

Soto-Crespo, J. M.; Akhmediev, Nail; Mejía-Cortés, Cristian; Devine, Natasha

doi:10.1364/oe.17.004236

Cited by 53 publications

(28 citation statements)

References 25 publications

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“…Dissipative vortex solitons have been found in laser amplifiers [4], in Bose-Einstein condensates [5,6], and in systems described by the complex cubic-quintic Ginzburg-Landau equation [7][8][9]. Vortex solitons in uniform dissipative media may experience considerable dynamical shape deformations, as it occurs in suitable GinzburgLandau systems [8,9] and in wide-aperture lasers with a saturable absorber [4].Recently it was predicted that the evolution of nonlinear excitations in dissipative medium is affected dramatically by a spatially modulated gain. Such evolution has been studied in Bragg gratings and optical waveguides [10,11], in materials with periodic refractive index modulation [12,13], and in Bose-Einstein condensates [14].…”

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confidence: 99%

Rotating vortex solitons supported by localized gain

et al. 2011

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We show that ring-like localized gain landscapes imprinted in focusing cubic (Kerr) nonlinear media with strong two-photon absorption support new types of stable higher-order vortex solitons containing multiple phase singularities nested inside a single core. The phase singularities are found to rotate around the center of the gain landscape, with the rotation period being determined by the strength of the gain and the nonlinear absorption.Vortex solitons are self-sustained excitations carrying a nonzero angular momentum and topological phase singularities [1]. In uniform focusing local media, vortex solitons exhibit ring-like profiles, a property that makes them prone to azimuthal modulation instabilities. Such instability can be suppressed in conservative materials by several mechanisms, including competing nonlinearities, external potentials, or nonlocalities. Stable vortex solitons form also in dissipative settings [2,3]. Dissipative vortex solitons have been found in laser amplifiers [4], in Bose-Einstein condensates [5,6], and in systems described by the complex cubic-quintic Ginzburg-Landau equation [7][8][9]. Vortex solitons in uniform dissipative media may experience considerable dynamical shape deformations, as it occurs in suitable GinzburgLandau systems [8,9] and in wide-aperture lasers with a saturable absorber [4].Recently it was predicted that the evolution of nonlinear excitations in dissipative medium is affected dramatically by a spatially modulated gain. Such evolution has been studied in Bragg gratings and optical waveguides [10,11], in materials with periodic refractive index modulation [12,13], and in Bose-Einstein condensates [14]. In such systems solitons form due to the interplay between localized gain and nonlinear absorption, and a spatial localization of the gain landscape ensures the stability of the background field. Such interplay may result in suppression of azimuthal modulation instabilities and formation of stable stationary vortex solitons [15]. However, the possibility to excite rotating vortex states in such systems has never been addressed to date.In this Letter we reveal that ring-like gain landscapes imprinted in cubic media with two-photon absorption support rich families of rotating vortex states exhibiting multiple phase singularities embedded into a single vortex core and possessing azimuthally modulated shapes. Notice that while in some conservative systems the formation of vortex solitons with multiple singularities may be stimulated by external factors, such as large sample asymmetry in nonlocal media [16] or the presence of strong optical lattices [17], in our dissipative setting the asymmetric rotating vortices appear despite the fact that all parameters are either spatially uniform (diffraction and nonlinearity) or radially symmetric (gain). This indicates that the formation of such states is mediated by complex internal energy flows that may be highly asymmetric in a system with localized gain.We consider the propagation of laser radiation along the  -axis in a...

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confidence: 99%

Rotating vortex solitons supported by localized gain

et al. 2011

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“…The first class of papers [14-16, 19, 22] used dynamical systems techniques to prove that CGLE admits periodic and quasi-periodic traveling wave solutions, while the second class of papers [6,9,10], primarily involving numerical simulations of the full CGLE in the context of nonlinear optics, revealed various branches of plane wave solutions which are referred to as continuous wave (CW) solutions. More importantly, these latter studies also found various spatially confined coherent structures with envelopes which exhibit complicated temporal dynamics [2,31,39,40]. All indications are that these classes of solutions, all of which have amplitudes that vary in time, do not exist as stable structures in Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 89%

Pulses and snakes in Ginzburg–Landau equation

Mancas

Choudhury

2014

Nonlinear Dyn

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Using a variational formulation for partial differential equations (PDEs) combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE are varied. Our predictions on the variation of the soliton amplitude, width, position, speed and phase of the solutions using the variational formulation agree with simulation results.First, we develop a variational formalism which explores the various classes of dissipative solitons. Given the complex dynamics the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple, and the trial function integrable while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses and snakes.In addition, the resulting Euler-Lagrange (EL) equations from the variational formulation are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons, we use dynamical systems theory to focus on more complex attractors viz. periodic (pulses) and quasiperiodic (snakes). Periodic evolution of the trial function parameters on stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent.Secondly, we investigate the dissipative solitons of the CGLE and analyze its qualitative behavior by using numerical methods for ODEs. To solve numerically the nonlinear systems of ODEs that represent EL equations obtained from variational technique, we use an explicit Runge-Kutta fourth order method (RK4).Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent due to the lack of dissipation.

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“…It shows a variety of coherent structures like stable and unstable pulses, fronts, sources and sinks in 1D, see [3,36,38,39], vortex solitons, see [13], spinning solitons, see [14], dissipative ring solitons, see [33], rotating spiral waves, propagating clusters, see [30], and exploding dissipative solitons, see [34] in 2D as well as scroll waves and spinning solitons in 3D, see [22]. We are interested in exponentially localized rotating wave solutions u : p .…”

Section: Rotating Waves In Reaction Diffusion Systemsmentioning

confidence: 99%

Spatial decay of rotating waves in reaction diffusion systems

Beyn

Otten

2016

Dynamics of Partial Differential Equations

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Abstract. In this paper we study nonlinear problems for Ornstein-Uhlenbeck operatorswhere the matrix A ∈ R N,N is diagonalizable and has eigenvalues with positive real part, the map f : R N → R N is sufficiently smooth and the matrix S ∈ R d,d in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution v of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that v belongs to an exponentially weighted Sobolev space W 1,p. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution v of the eigenvalue problemdecays exponentially in space, provided Re λ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.

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Dissipative ring solitons with vorticity

Cited by 53 publications

References 25 publications

Rotating vortex solitons supported by localized gain

Rotating vortex solitons supported by localized gain

Pulses and snakes in Ginzburg–Landau equation

Spatial decay of rotating waves in reaction diffusion systems

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