Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity

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Stability limits for three-dimensional vortex solitons in the Ginzburg-Landau equation with the cubic-quintic nonlinearity

D. Mihalache,¹D. Mazilu,¹F. Lederer,²H. Leblond,³ and B. A. Malomed⁴

1Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 407 Atomistilor, Magurele-Bucharest, 077125, Romania

2Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller Universität Jena, Max-Wien-Platz 1, D-077743 Jena, Germany

3Laboratoire POMA, UMR 6136, Université d’Angers, 2 Bd Lavoisier, 49000 Angers, France

4Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 共Received 26 July 2007; revised manuscript received 3 October 2007; published 23 October 2007兲 We complete the stability analysis for three-dimensional dissipative solitons with intrinsic vorticitySin the complex Ginzburg-Landau equation with cubic and quintic terms in its dissipative and conservative parts. It is found and qualitatively explained that a necessary stability condition for all vortex solitons, but not for the fundamental ones 共S= 0兲, is the presence of nonzero diffusivity in the transverse plane. The fundamental solitons are stable in all cases when they exist, while the vortex solitons are stable only in a part of their existence domain. However, the spectral filtering共i.e., the temporal-domain diffusivity兲is not necessary for the stability of any species of dissipative solitons. In addition to the recently studied solitons with S= 0 , 1 , 2, a stability region is also found for ones withS= 3.

DOI:10.1103/PhysRevA.76.045803 PACS number共s兲: 42.65.Tg, 42.65.Sf, 47.20.Ky

The complex Ginzburg-Landau共CGL兲equation provides a basic mathematical framework for studies of dissipative dynamics in one- and multiple-dimensional geometries in fluid flows, plasmas, chemically reacting mixtures, nonlinear optical cavities and lasers, and other physical media; see re- views in 关1兴 and references therein. A significant class of patterns generated in the framework of the CGL equation are solitary 共localized兲 pulses, alias dissipative solitons 共DSs兲关2–4兴. The search forstablelocalized states is a challenge in the two- and three-dimensional 共2D and 3D兲 geometries, where the instabilities known in 1D media, such as the blowup of the zero background due to the presence of a linear gain in the CGL equation, are combined with the pos- sibility of critical 共in 2D兲 or supercritical 共in 3D兲 collapse induced by cubic self-focusing nonlinearity and the vulner- ability ofvortex solitons 共alias vortex tori in 3D兲—i.e., localized patterns with embedded vorticity—to azimuthal perturbations that tend to split them关5,6兴.

A model that may support stable localized patterns in any dimension is provided by the CGL equation with cubic- quintic共CQ兲nonlinearity. This equation was originally intro- duced in 2D form by Petviashvili and Sergeev 关7兴. Then, solitary pulses were studied in detail in its 1D version关8兴. In the 2D geometry, stable vortical共alias spiral兲 DSs, with the vorticity共“spin”兲S= 1 and 2, were found, in the framework of the CQ CGL equation, in Ref. 关9兴. Stable fundamental, alias spinless共S= 0兲, 3D spatiotemporal solitons关10兴, as well as double-soliton complexes, including rotating ones 关11兴, have been found in optical models based on the CQ CGL equation. In recent works关12,13兴, 3D fundamental and spin- ning DSs with S= 1 and 2 were constructed by means of numerical methods and their stability was investigated through computation of the growth-rate eigenvalues for perturbation eigenmodes.

We take the 3D CGL equation with the CQ nonlinearity in the following form:

iU_z+

冉

¹²⁻ⁱ^␤

冊

^共^U^xx⁺^U^yy^兲⁺

冉

^D² ⁻ⁱ^␥

冊

^U^tt

+关i␦⁺共1 −i␧兲兩U兩²−共␯⁻ⁱ␮兲兩U兩⁴兴U= 0. 共1兲

In applications to optics, U is the local amplitude of the electromagnetic wave in the bulk medium, propagating along thezaxis, the temporal variable beingt=T−z/V₀, whereTis time andV₀the group velocity of the carrier wave. Here, the coefficients that are scaled to be 1 / 2 and 1 account, respectively, for the diffraction in the transverse plane共x,y兲and the self-focusing Kerr nonlinearity,␤ⱖ0 is the effective diffu- sivityin the transverse plane, positive real constants␦^,␧, and

␮represent, respectively, linear loss, nonlinear共cubic兲gain, and nonlinear 共quintic兲 loss, which are basic ingredients of the CQ CGL equation 关7兴, ␯ⱖ0 accounts for the self- defocusing quintic correction to the Kerr term共saturation of the optical nonlinearity兲, Dis the group-velocity dispersion 共GVD兲 coefficient 共D⬍0 and D⬎0 correspond to the normal and anomalous GVD, respectively兲, and ␥⬎0 is its counterpart accounting for the spectral filtering—i.e., the dispersion of the linear loss.

While the physical meaning of other parameters is quite clear, in terms of the standard optical model, the spatial diffusivity 共⬃␤兲 requires a special discussion. This term is known, in particular, in models of large-aspect-ratio lasers, just above the first lasing threshold. The general model is based on the complex Swift-Hohenberg equation关14兴, which reduces to the CGL equation if the fourth-order spatial de- rivative is neglected. The spatial diffusivity derived in the latter case is ␤^{= −}␶p␶c⌬共␶p+␶c兲⁻², where ␶p is the polarization-dephasing time,␶cthe cavity-decay time, and⌬ detuning between the cavity’s and atomic frequencies. There- fore, the relevant case of ␤⬎0 corresponds to a negative detuning. In the same laser-cavity model, the spectral filtering is mainly related to the gain bandwidth ␻g, being ␥ PHYSICAL REVIEW A76, 045803共2007兲

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=g/共兩␤2兩␻g

2兲, wheregis the gain and␤2the GVD coefficient in physical units.

We note that, in the conservative counterpart of the CQ CGL equation—i.e., the nonlinear Schrödinger共NLS兲 equation with CQ nonlinearity—the quintic term must be self- defocusing共in the 2D and 3D settings兲 to suppress the collapse driven by the self-focusing cubic nonlinearity关5,15兴. In Ref. 关12兴, it was shown that a self-defocusing sign of the quintic term is not necessary for the stability of 3D fundamental and vortex DSs, as collapse is prevented by the action of the quintic term in the dissipative part of the equation. As concerns the role of the GVD coefficient 共D兲, in Ref. 关12兴 stable fundamental and vortical 3D solitons were found in the model with anomalous dispersion, D⬎0, while in Ref.

关13兴they were also found in the normal-GVD model, with D⬍0, both cases being physically relevant. It was concluded that␤⬎0 is a necessary condition for the stability of vortex solitons in the 3D model, while fundamental DSs may also be stable for ␤^{= 0} 关13兴. However, all the results were re- ported for ␥⬎0 and only for the 3D solitons with S

= 0 , 1 , 2.

The objective of this work is to demonstrate that, while

␤⬎0 共nonzero diffusivity in the transverse plane兲 is indeed necessary for the stability of all vortical DSs, the presence of spectral filtering is not a necessary stability condition; i.e., stable 3D dissipative solitons, both fundamental and vortical ones, exist at␥= 0, which opens the way to study collisions between the solitons moving in the longitudinal direction.

We will also demonstrate that the stability is not limited to the vortex DSs withSⱕ2, as families of stable solitons with S= 3 can be readily found too.

We look for stationary solutions to Eq.共1兲with vorticityS as U共z,x,y,t兲=␺共r,t兲exp共ikz+iS␪兲, where r and ␪ ^{are the} polar coordinates in plane共x,y兲,kis the wave number, and the function␺共r,t兲obeys the stationary equation

冉

¹²⁻ⁱ^␤

冊冉

^␺^rr⁺¹^r^␺^r⁻^S^r²²^␺

冊

⁺

冉

¹²^D⁻ⁱ^␥

冊

^␺^tt

+关i␦⁺共1 −i␧兲兩␺兩²−共␯⁻ⁱ␮兲兩␺兩⁴兴␺⁼^k␺^. 共2兲 Localized solutions to this equation must decay exponen- tially atr,兩t兩→⬁, and asr^S atr→0 共assumingSⱖ0兲.

To find the stationary DSs, we simulated the propagation of localized vortices or zero-vorticity pulses within the framework of the radial version of Eq.共1兲, obtained by the substitution of U共z,x,y,t兲=⌿共z,r,t兲exp共iS␪兲 and starting with input pulses corresponding to the given vorticity,

⌿共0 ,r,t兲=Ar^Sexp关−共r²/␳²⁺^t²^/␶²兲/ 2兴, with some constants A,␳^{, and}␶. The standard Crank-Nicholson scheme was used for the numerical integration共typically, with transverse and longitudinal step sizes⌬r=⌬t= 0.2 and⌬z= 0.01兲. However, this numerical procedure does not guarantee true stability of the vortex states, as they may be unstable against azimuthal perturbations关5兴. The full stability was explored in a different way; see below.

Results are presented below for GVD coefficient D= 1 共anomalous dispersion兲, a fixed coefficient of the self- defocusing quintic correction to the Kerr nonlinearity,

␯= 0.1, linear loss ␦^{= 0.4, and} zero spectral filtering, ␥^{= 0.}

The latter choice is the most essential difference from the analysis performed in Refs.关12,13兴. The other parameters of Eq. 共1兲—viz., diffusivity ␤, cubic gain ␧, and quintic loss

␮—were varied. Families of the 3D fundamental and vortex solitons, withS= 0 , 1 , 2 , 3, are represented in Figs. 1共a兲and

1共b兲 by dependences of the energy, E

⬅2␲兰₀^⬁rdr兰_−⬁^+⬁dt兩␺共r,t兲兩², and propagation constant k on␧, at␤^{= 0.5}关the curvek=k共␧兲forS= 3 completely overlaps in Fig. 1共b兲 with its counterpart for S= 2兴. Figure 1 includes results of the stability analysis for the DSs; see below.

Typical radial and temporal cross sections of the stable DSs with S= 0 , 1 , 2, and S= 3 are shown in Figs. 2共a兲–2共d兲 for␤^{= 0.5,}␮^{= 1, and}⑀^{= 2.2}共the radial and temporal shapes FIG. 1. The energy共a兲and wave number共b兲of fundamental and vortical dissipative solitons in the 3D complex Ginzburg-Landau equation versus the cubic-gain parameter␧for fixed diffusivity,␤

= 0.5, and quintic loss,␮= 1. Here dotted and solid lines represent unstable and stable solutions, respectively. Arrows indicate stability borders.

FIG. 2. Cross-section shapes of typical stable 3D dissipative solitons, with S= 0 , 1 , 2 , 3, in the transverse 共r兲 and temporal 共t兲 directions, for␤= 0.5,␮= 1, and⑀= 2.2关in共b兲, the temporal shapes forS= 2 andS= 3 completely overlap with that forS= 1; real and imaginary parts ␺r and ␺iof the temporal shapes for S= 2 and S

= 3 are displayed in panels共c兲and共d兲兴.

BRIEF REPORTS PHYSICAL REVIEW A76, 045803共2007兲

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are shown, respectively, acrosst= 0 andr=r_max, where local power 兩␺共r,t兲兩² attains its maximum兲. Note that, while the temporal shapes of the solitons withS= 2 and 3 are virtually identical in terms of兩␺兩, they are quite different as concerns their real and imaginary parts; see panels2共c兲and2共d兲.

As the next step, we have performed the computation of instability growth rates for eigenmodes of small perturbations. To this end, a perturbed solution to Eq.共1兲was looked for as

U=关␺共r,t兲+f共r,t兲exp共␭z+iJ␪兲

+g^*共r,t兲exp共␭^*z−iJ␪兲兴exp共ikz+iS␪兲, 共3兲 where integerJand共complex兲␭are the azimuthal index and growth rate of infinitesimal perturbations represented by eigenmodes f andg. The substitution of this expression in Eq. 共1兲 leads to linearized equations which were solved by means of the same numerical methods as in Refs.关12,13兴.

The results thus obtained for stability are incorporated in Fig.

1. Further, in Fig. 3 we display the existence and stability domains for the fundamental共S= 0兲 and vortical共S= 1 , 2 , 3兲 DSs in the plane of共␮,␧兲for a nonzero diffusivity,␤^{= 0.5.}

The fundamental DSs are stable in their entire existence domain—i.e., between the upper and lower lines in Fig.

1共a兲—whereas the vortex solitons with S= 1 , 2 , 3 exist between borders ␧=␧upp共␮兲 and ␧=␧low共␮兲, being stable in narrowerregions, between dotted curves in panels共b兲–共d兲of Fig.3. Direct simulations demonstrate that, beneath curves

␧=␧low共␮兲, all input pulses decay, whereas above the upper borders,␧=␧_upp共␮兲, they expand indefinitely in the temporal

direction, remaining localized in the spatial domain.

As mentioned above, and stated in Ref.关13兴 for the set- ting with␥⬎0, the transverse diffusivity,␤⬎0, is necessary for the stability of all vortex DSs, but not of their fundamental counterparts. We have checked that this conclusion re- mains true in the model with zero spectral filtering,␥^{= 0. In} particular, if ␤= 0, the perturbation eigenmode with azimuthal indexJ= 2关see Eq.共3兲兴destabilizes the family of the S= 1 vortex DSs in theentire region of their existence. The crucial role of the diffusivity in the stabilization of the vortex DSs can be understood. Indeed, as mentioned above, a well- known fact is that the most dangerous instability mode for vortex solitons is an azimuthal one, which tends to split them into several separating fragments关5,6兴(if the azimuthal in- dex of the dominant unstable mode is J 关see Eq. 共3兲兴, the number of the breakup splinters will also beJ). On the other hand, the diffusivity term in Eq.共1兲 induces a brake force which suppresses the free motion of the splinters; hence, it helps to prevent the splitting instability.

In the case when the diffusivity is present 共␤= 0.5兲, the results of the linear stability analysis for the vortex DSs with S= 3 are displayed in Fig.4, where, fixing␮= 1, we display the largest instability growth rate versus nonlinear gain␧共the stability region revealed by this figure is included in Fig.3兲.

Note that the actual borders of the stability regions for the vortex solitons with vorticitySare determined by the pertur- FIG. 3. The existence and stability domains of 3D dissipative

solitons, withS= 0 , 1 , 2 , 3, for ␤= 0.5, in the plane of the quintic- loss and cubic-gain coefficients共␮,␧兲. The fundamental共S= 0兲soli- tons exist and are stable between solid curves. The vortex solitons withS= 1 , 2 , 3 exist between solid curves and are stable between dotted ones; i.e., their stability domain issmallerthan the existence range.

FIG. 4. The largest instability growth rate versus ␧ for S= 3 vortex solitons. Other parameters are␮= 1,␤= 0.5, and␥= 0. The arrow indicates the center of the stability interval.

FIG. 5. The recovery of a perturbed 3D vortex soliton withS

= 3, for␮= 1, ␤= 0.5, and ␧= 2.3. Upper row: the intensity 共兩U兩²兲 and phase of the initial soliton perturbed by random noise. Lower row: the same in the self-cleaned soliton atz= 800. The simulations were run on a grid of size关−14, 14兴⫻关−20, 20兴⫻关−20, 20兴.

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bation eigenmodes withJ=S共see Fig.4for theS= 3 case兲. It is plausible that stability regions can be found for␤⬎0共and

␥ⱖ0兲also forS⬎3.

In fact, the stable 3D DSs are strong attractors: direct simulations demonstrate that an arbitrary initial pulse with intrinsic vorticity S readily converges to the respective DS.

Accordingly, the DSs which are predicted to be stable by linear analysis turn out to be robust against the addition of nonsmall initial perturbations. The robustness of the DSs was tested, in particular, by taking the perturbed state as U共z

= 0兲=␺共r,t兲共1 +q␾兲exp共iS␪兲, where q is a perturbation amplitude and ␾ a random variable uniformly distributed in interval关−0.5, 0.5兴. The simulations reveal that those vortex solitons which were predicted to be unstable by the linear analysis either completely decay, or expand indefinitely along the temporal direction 共remaining localized in the transverse plane兲. On the other hand, the linearly stable DSs are indeed robust, as shown in Fig.5 for the vortex soliton

withS= 3, the initial perturbation amplitude taken at the level of 10%.

In this work we aimed to complete the analysis of condi- tions for the existence and stability of 3D DSs with intrinsic vorticity S in the cubic-quintic complex Ginzburg-Landau equation. Essential conclusions are that nonzero diffusivity in the transverse plane共␤⬎0兲 is necessary for the stability of all vortex DSs, but not for the fundamental ones共S= 0兲. At

␤ⱖ0, the fundamental DSs are stable in their entire existence area, while the vortex solitons are stable, at␤⬎0, only in a part of their existence range. On the other hand, the presence of the spectral filtering is not necessary for the stability of any type of the DSs. In addition to the DSs withS

= 0 , 1 , 2 considered in Refs. 关12,13兴, we have also found a stability area for their counterparts withS= 3.

This works was supported, in part, by the Deutsche Forschungsgemeinschaft共DFG兲, Bonn.

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