Dark-type solitons in media with competingnonlocal non-Kerr nonlinearities

Zhongxiang Zhou,* Yanwei Du, Chunfeng Hou, Hao Tian, and Ying Wang

Department of Physics, Harbin Institute of Technology, Harbin, 150001, China*Corresponding author: zhouzx@hit.edu.cn

Received February 2, 2011; revised April 2, 2011; accepted April 28, 2011;posted April 28, 2011 (Doc. ID 142165); published May 31, 2011

We address the properties of dark-type solitons, including dark solitons and darklike bright solitons, based on aphenomenological model for a nonlocalmedium featuring competing cubic-quintic nonlinearities [Phys. Rev. E 7466614 (2006)]. We consider two forms of such nonlinearities: focusing cubic and defocusing quintic nonlinearities,and defocusing cubic and focusing quintic nonlinearities. We reveal that nonlocality drastically modifies shapes,velocity, existence properties, and stability properties of dark-type solitons. At suitable parameter regions, non-locality could impose strong restrictions on soliton existence or exhibit remarkable destabilizing action on dark-type solitons. The stability of single dark-type solitons exactly obeys the stability criterion for dark solitons. Colli-sions between dark-type solitons and their bound states are also investigated. © 2011 Optical Societyof America

OCIS codes: 190.0190, 190.6135.

1. INTRODUCTIONIn recent years, much attention has been attracted to the studyof nonlocal solitons since the pioneering work of Snyder andMitchell [1]. In contrast to the local response, in nonlocal med-ia, the refractive index of one point depends on the beam in-tensity in a certain neighborhood of this point. Nonlocalityof nonlinear response has been observed in many opticalmaterials, such as nematic liquid crystals, thermal-opticalmaterials, plasmas, and photorefractive crystals. In addition,it has been addressed that solitons in quadratic nonlinearmaterials are equivalent to nonlocal solitons [2]. In thiscontext, accurate determination of the limit for good pulse-compression becomes feasible by analyzing the simple nonlo-cal nonlinear models [3]. Nonlocality of nonlinear responsessuppresses the modulational instability of plane waves [4,5],arrests the collapse of multidimensional solitons [6], and al-lows the existence of bound states [7,8] (see a review in[9]). Moreover, in the two-dimension setting, extensive studieshave also shown that nonlocality has a profoundly stabilizingaction on solitons featuring complex structures, such as multi-pole solitons [10], ring vortex solitons [11–13], soliton clusters[14], spiraling solitons [15–17], X-waves [18], gap solitons [19],and incoherent solitons [20]. However, most of the aforemen-tioned studies concentrate on the simplest model of nonlocalKerr response, which is ubiquitous in descriptions of impor-tant nonlocal media such as lead glasses [21] and nematicliquid crystals [22]. There are also nonlocal media whose non-linear responses should take into account potential satura-tions of the nonlocal nonlinearity, such as atomic vapors[23–25]. Particularly, Mihalache et al.introduced a new pheno-menological model for nonlocal media featuring cubic-quintic(CQ) nonlocal nonlinearities [26]. Since the CQ nonlinearitycan be regarded as a power-law expansion for saturable non-linearity, it can serve as an approximate model describing

beam propagation in nonlocal media with a saturation ofthe nonlinear response.

Dark solitons are localized light intensity dips with antisym-metric phase profiles propagating in the plane-wave (nonvan-ishing) background [27]. Like their bright counterparts, theyare the fundamental solutions of the nonlinear Schrödingerequation with defocusing nonlinearity. In addition to the con-ventional Kerr medium, the study of dark solitons alsoextends to CQ and saturable media that offer greater flexibil-ity for potential device designs [28–34]. More interestingly, un-der suitable conditions, such media support a novel class ofdarklike bright solitons that have no counterparts in defocus-ing Kerr media [35]. They are bright solitons with nonvanish-ing flat tails and have antisymmetric phase profiles similar tothose of dark solitons. During recent years, dark solitons havealso been the subject of extensive research in nonlocal Kerrmedia [8,36–41]. In typical nonlocal media, the attraction ofdark solitons and the formation of bound states that are insharp contrast to local Kerr media have been demonstratedboth theoretically [8] and experimentally [37]. However, fordark solitons, few works have taken into account both satura-tion and nonlocality of nonlinear response; a very recent workby Tsoy focuses on the weakly nonlocal case [42]. The presentpaper focuses on the properties of dark-type solitons in com-peting CQ media with an arbitrary degree of nonlocality,which may shed light on the effect of the interplay betweensaturation and nonlocality.

In this paper, we introduce properties of dark-type solitonsbased on Mihalache et al.’s phenomenological model for non-local media [26] featuring competing cubic-quintic CQ nonli-nearities. We show that nonlocality drastically modifies shape,velocity, existence properties, and stability properties of dark-type solitons. We dicuss our finding that dark-type solitonsstrictly follow the stability criterion of dark solitons and

Zhou et al. Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. B 1583

0740-3224/11/061583-08$15.00/0 © 2011 Optical Society of America

can form stable bound states. Moreover, interesting solitoncollision scenarios are displayed.

2. PROPAGATION MODEL ANDMODULATIONAL INSTABILITYWe consider the light propagation in nonlocal media withcompeting CQ nonlinearities, described by the followingequations for dimensionless amplitude of light field ψ and con-tributions to the refractive index n3 and n5

i∂ψ∂ξ þ 1

2∂2ψ∂η2 þ ðn3 þ n5Þψ ¼ 0; n3 − σ3

∂2n3

∂η2 ¼ α3jψ j2;

n5 − σ5∂2n5

∂η2 ¼ α5jψ j4; ð1Þ

where η and ξ are the transverse and longitudinal coordinatesscaled to the characteristic width of the dark-type solitons andto the diffraction length, respectively. Parameters σ3;5 denotethe corresponding nonlocal lengths of the cubic and quinticnonlinearities. The case σ3;5 → 0 corresponds to the localCQ medium, while σ3;5 → ∞ relates to a strong nonlocality.Note that the nonlinear contribution to refractive index inEq. (1) can also be represented as n3;5 ¼

R∞−∞

G3;5ðη−η0Þjψðη0 − ξÞj3;5dη0, where G3;5ðηÞ ¼ ð1=2 ffiffiffiffiffiffiffiffiσ3;5

pexpð−jηj=ffiffiffiffiffiffiffiffiσ3;5

p ÞÞ are the response functions of the nonlocal medium.Parameters α3 and α5 characterize the strength and sign ofthe cubic and quintic nonlinearities [positive (negative) valuescorrespond to focusing (defocusing)]. Here, we consider thetwo forms of the competition between the cubic and quinticnonlinearities: (I) α3 < 0, α5 > 0; and (II) α3 > 0, α5 < 0.

Note that under the coordinate transformation ψ̂ ¼ χψ ,η̂ ¼ η=χ, η̂0 ¼ η0=χ, ξ̂ ¼ ξ=χ2, σ̂3;5 ¼ σ3;5χ2, α̂3 ¼ α3,α̂5 ¼ α5=χ2, ψ , and ψ̂ satisfy the same evolutional equationin Eq. (1). Owing to this coordinate transformation, thestrength of the cubic nonlinearity and the background inten-sity of the dark soliton may be set jα3j≡ 1 and jψð�∞; ξÞj → 1,respectively. At the same time, we could consider darksolitons with different background intensity jψð�∞; ξÞj byvariations of α5 and σ3;5.

The stationary dark-type solitons solutions could be consid-ered in the moving coordinates ξ, η ¼ η − vξ with the formψðη; ξÞ ¼ ρðηÞ exp½iθðηÞ þ ibξ�. Here, ρ and θ represent thefield amplitude and phase distribution, and b is the propaga-tion constant. Inserting such expressions into Eq. (1) andaccounting for the asymptotic behaviors, jψð�∞; ξÞj → 1and b ¼ α3 þ α5, yields the equations

dθdη ¼ v

�1 −

1

ρ2�; ð2Þ

12d2ρdη2 þ

v2

2

�ρ − 1

ρ3�þ ðn − bÞφ ¼ 0; ð3Þ

where n ¼ α3R∞−∞

R3ðξ − ξ0Þρðξ0Þ2dξ0 þ α5R∞−∞

R5ðξ −ξ0Þρðξ0Þ4dξ0 is the induced refractive index. In addition, forfurther use, note that Eq. (1) conserves several importantrenormalized invariants, including the renormalized energyflow Pr and momentum Mr

Pr ¼Z

∞

−∞

ðjψ j2 − 1Þdξ ¼Z

∞

−∞

ðρ2 − 1Þdξ; ð4Þ

Mr ¼i2

Z∞

−∞

ðψ�ξψ − ψξψ�Þð1 − 1=jψ j2Þdξ ¼ −v

Z∞

−∞

ðρ2 − 1Þ2ρ2 dξ

ð5Þ

Moreover, the soliton grayness, which strongly depends onsoliton velocity v, is defined as g ¼ minðρ2Þ [g ¼ maxðρ2Þ fordarklike bright solitons]. g and Pr then might serve as a mea-surement for soliton shapes. It is well known that in localmedia, the sign for dMr=dv decides the stability propertiesof dark-type solitons: positive (negative) corresponds tostable (unstable) solitons [27]. It may also be considered asan indication of stability for dark-type solitons in non-local media, although the stability theory there has not beenestablished.

At the local limit, σ3;5 ¼ 0, Eq. (3) recovers to the usual localCQ model, which has the the following analytical solutions[29,33,35]

ρ2ðξÞ ¼ 1 −2k2

a� b coshð2kξÞa ¼ 8α5=3 − α3; b ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ 4α5k2=3

q; ð6Þ

where the soliton amplitude k is defined by the soliton velocityv through the relation k2 þ 2v2 ¼ 4α5 − 2α3. In CQ model (I),Eq. (6) describes two different dark-type solitons, includingthe first solution, with the sign þ corresponding to a standarddark soliton, and the second one, with sign − for a darklikebright soliton. In contrast, in CQ model (II), Eq. (6) permitsonly the standard dark soliton solution. In all cases, thedark-type solitons come to exist provided that the plane-wavebackgrounds are modulationally stable α3 þ 2α5 < 0.

For the nonlocal model we consider here, it is also instruc-tive to consider first the modulational instability of the planewaves solution of Eq. (1). Following the procedure listed in[4], we obtained the following equation for the stability ofthe plane-wave ψ ¼ ρ0 exp½iðk0x − ω0zÞ�:

λ2 ¼ −k̂2�k̂2=4 −

α31þ σ3k̂2

− 2α5

1þ σ5k̂2�: ð7Þ

Here, λ and k̂ denote the perturbation growth rate of the planewave upon propagation and the spatial frequency, respec-tively. It is imaginary when the plane wave becomes modula-tionally unstable. The precondition for the existence of darksolitons is that the background plane wave must be modula-tionally stable (λ is real). It is easy to see that, like the darksoliton in local CQ medium, the inequality, α3 þ 2α5 < 0, guar-antees stable plane-wave solutions for nonlocal media withequal nonlocal lengths σ3 and σ5. When the two nonlinearitycomponents have unequal nonlocal lengths (a situation thatalso can be encountered in media featuring synthesis of non-local nonlinearities [43,44]), there are additional restrictionsfor the stability of plane wave. Figures 1(a) and 1(b) displayseveral stability parameter regions for plane waves at fixed σ3for CQmodels (I), and σ5 for (II), respectively. It can be clearlyseen that the instability regions expand as the nonlocal length

1584 J. Opt. Soc. Am. B / Vol. 28, No. 6 / June 2011 Zhou et al.

of the corresponding defocusing nonlinearity component in-creases. This is consistent with the general concept: the morenonlocal the nonlinearity is, the weaker it is. Note that such aconcept can also explain the decline of maximum velocity ofdark solitons, that usually depends on the strength of the non-linearity, in nonlocal Kerr media [38]. Also, at fixed α3;5 and σ3for CQmodel (I) [σ5 for CQmodel (II)], it can be expected thatthe maximum velocity of the dark-type solitons is a monoto-nically increasing function of σ5 (σ3). Such a behav-ior is shown in Figs. 2(e), 3(d), and 4(e).

3. STATIONARY SOLUTIONS AND THEIRSTABILITYLocalized soliton solutions are achieved by solving Eq. (3) nu-merically with the Newton iteration method. Then Eq. (2) canbe easily integrated to obtain the phase distributions θ. In ad-dition, to study the stability and propagation dynamics ofdark-type solitons, through combinations of two remote iden-

tical solitons with opposite signs of phase distributions, weperformed numerical simulations of Eq. (1) using a split-stepFourier method.

Properties of standard dark solitons in CQ model (I) aresummarized in Fig. 2. The generic properties of such solitonsare reminiscent of those of solitons in nonlocal Kerr media[38]. Nonlocality of the nonlinear response strongly affectsthe shapes and maximal velocity of dark solitons. The profilesof intensity and relative phase distribution for dark solitonsfeature multiple intensity oscillations around the main inten-sity deep [Fig. 2(a)]. Similar to their local counterparts, Pr

(note that the form defined here is different from that in[38]) and g are still monotonically decreasing functions of v[Fig. 2(d) and 2(e)]. Figures 2(b) and 2(c) display the solitonprofiles and relative refractive index distributions for σ3 ≠ σ5.Such a nonlocal length mismatch leads to pronouncedvariations of soliton shape and maximal soliton velocity[Fig. 2(e)], particularly when σ3 > σ5. The renormalized

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

5

σ3=10

2

|α5|

σ 5

0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

|α5|

σ 3

5

2

σ5=10

Fig. 1. (Color online) Plane waves are modulationally stable above the curves in the two parameter planes. (a) and (b) correspond to competingnonlocal CQ models (I) and (II), respectively.

−25 0 250

0.9

1.8

η

ρ2

0.3

1.2

−n

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0.9

1.8

η

ρ2

0.3

1.2

−n

−25 0 250

0.9

1.8

η

ρ2

0.3

1.2

−n

0 0.2 0.4 0.6 0.80

1.3

2.6

v

Pr

10

5

σ3,5=0

0 0.2 0.4 0.60

1

v

g 105

σ5=0.8

0 5 10 15 20 250.32

0.38

0.44

0.5

σ3,5

α 5m

Fig. 2. (Color online) Intensity and relative refractive index distributions for single dark solitons in nonlocal CQ model (I) with (a) equal and (b)–(c) unequal nonlocal lengths σ3 and σ5. (d) Renormalized energy flow, and (e) darkness, versus velocity. (f) Existence regions for different nonlocallengths. Points marked by squares in (d) and (e) correspond to the solitons depicted in (a)–(c). Other parameters are α5 ¼ 0:2 for (a)–(e); in (e),σ3 ¼ 5.

Zhou et al. Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. B 1585

momentum Mr is the monotonically increasing function of v,which indicates that the dark solitons may be stable. Throughintensive numerical simulations, we checked this and foundthat the dark solitons are stable, although at high soliton ve-locities, they may exhibit weak instability on their surround-ing tails [Fig. 5(a)].

Particularly in strong nonlocal media, there is a maximalstrength of focusing quintic component αm5 above which darksolitons cannot exist, although the plane waves are modula-tionally stable [Fig. 2(f)]. This is in sharp contrast with localmedia, where dark solitons come to exist even near the mod-ulational instability limit α5 → 0:5. Therefore, nonlocality ofthe nonlinear response imposes drastic restrictions on theexistence of dark solitons.

Figure 3 displays properties of darklike bright solitons un-der the same model. We can see that the nonmonotonic tailsfor such solitons are not as pronounced as for dark solitons[Figs. 3(a) and 3(b)]. Nonlocality leads to shrink for both thepeak intensity amplitude [Fig. 3(c)] and the maximal solitonvelocity vm [Fig. 3(f)]. This is more pronounced when α5 issmall. At small and moderate velocities, the solitons’ inducedrefractive index distribution exhibits typical W-type profiles[Fig. 3(a)], similar to their counterparts in the local CQ med-ium. At high velocity, it may transform into a single broadwaveguide [Fig. 3(b)], and the solitons then can be stable. Re-call that in the local CQ medium, the stable solitons can existonly for α5 >¼ αcr5 ¼ 0:3276. Figures 3(d) and 3(e) display therenormalized momentum Mr versus velocity at α5 ¼ 0:33 andα5 ¼ 0:35. We see that Mr becomes a monotonic decreasingfunction of v at σ3;5 ¼ 5 [Fig. 3(d)] and σ3;5 ¼ 10 [Fig. 3(e)].This indicates the shrink of the stability regions. It can be seenmore clearly in Fig. 3(f) that the stability region for darklikesolitons exists only for α5 >¼ αcr5 ¼ 0:34. At higher nonlocallengths σ3:5 ¼ 10, αcr5 actually increases, to 0.36. Therefore,nonlocality has a destabilizing action on darklike bright soli-

tons, which is in contrast to the general concept: nonlocalitystabilizes solitons. However, it should be mentioned that theconcept may fail in several cases. As discussed in [4,5], de-pending on the profile of the response function, nonlocalitymay lead to modulational instability of plane waves in defo-cusing media. It was shown in [45] that a competing secondtype of nonlinearity can even allow existence of bright soli-tons (i.e., existence of modulational instability) in a defocus-ing Kerr medium. Moreover, the investigation in [10] indicatesthat necklace solitons can be destabilized by nonlocality.

When the nonlocal lengths for the cubic and quintic nonli-nearity component are not equal, depending on the specificvalues of nonlocal lengths σ3;5, the darklike bright solitonsmay induce different refractive index distributions than darksolitons. Moreover, the mismatch of nonlocal lengths resultsin the existence of stable solitons (note that this can occur inboth cases, σ3 > σ5 and σ3 < σ5) [Fig. 3(d)].

The simulations also confirm the above results from the sta-bility criterion. For example, at α5 ¼ 0:4, the critical solitonvelocity separating unstable and stable solitons is locatedat vcr ¼ 0:2543. Figures 5(b) and 5(c) display the propagationexamples for darklike bright solitons with the velocity close tovcr. It can be seen that under white-noise perturbations, theunstable soliton at v ¼ 0:25 suffers a catastrophic collapseupon propagation, while the stable soliton at v ¼ 0:26 isrobust enough to retain its structure over long distances.

We also consider the dark solitons in CQ model (II), whoseproperties are summarized in Fig. 4. Similar to the darklikebright solitons analyzed above, such solitons feature weaklynonmonotonic tails [Figs. 4(a) and 4(b)], except for the casev → vm. The darkness g increases while the renormalized en-ergy flow decreases with v. To shed more light on the effectthat nonlocality exerts on the dark solitons, we concentrateon soliton properties at α5 ¼ −0:75, which serves as a criticalpoint for the local medium [Fig. 4(f)]. We can see that the

−30 0 300.7

1.7

η

ρ2

−0.63

−0.53

n

−30 0 300.7

1.7

η

ρ2

−0.63

−0.53

n

0 0.2 0.4 0.61.5

2.6

v

g

σ3,5=0

510

0 0.2 0.4 0.6−1.3

0

v

Mr

σ3,5=(0,0)

(5,5)(10,10)

(5,3)

(5,10)

0 0.28 0.56−1

0

v

Mr

σ3,5=0

5

10

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

α5

v

σ3,5=5

0

Fig. 3. (Color online) Intensity and relative refractive index distributions for single darklike bright solitons at (a) v ¼ 0:15 and (b) v ¼ 0:3 in thenonlocal CQ model (I) with α5 ¼ 0:4 and σ3;5, respectively. (c) Darkness, and (d) renormalized momentum, versus velocity at α5 ¼ 0:33. (e) Re-normalized momentumMr versus velocity at α5 ¼ 0:35. (f) Existence (below the squares) and stability regions (above the circles) of darklike brightsolitons. The blue dashed curves in (c)–(f) correspond to nonlinear media at the local limit (σ3;5 ¼ 0).

1586 J. Opt. Soc. Am. B / Vol. 28, No. 6 / June 2011 Zhou et al.

nonlocality drastically changes the soliton properties. Thedark soliton at v → 0 then is not a black soliton [Fig. 4(c)],while the maximal velocity decreases with σ3;5. The renorma-lized energy flow decreases faster in high nonlocal media[Fig. 4(d)]. In particular, the slope of the renormalizedmomentum becomes negative at low velocities [Fig. 4(e)],indicating that an instability region arises. The solitons’ in-duced refractive index distribution also explains this instabil-ity: at low velocity, it exhibits an M-type profile that isresponsive for the instability; as the velocity grows, it be-comes a single waveguide where stable propagating solitonscan be achieved. The computed instability regions (based onthe stability criterion) depicted in Fig. 4(f) give a more clearillustration of the effects caused by the nonlocality. The non-locality has a remarkable destabilizing action on dark solitons.We can see that the instability regions expand to α5 ≈ 0:82.Nevertheless, when there exists a mismatch of nonlocallengths, we can see that the dark solitons can be either un-stable or completely stable, depending on the specific valuesof nonlocal lengths σ3 and σ5; this is different than the darklikesolitons in CQ model (I).

Our simulations also confirm the above results from stabi-lity analysis. Representative propagation examples are dis-played in Figs. 5(d)–5(f). The critical velocity for CQ model(II) with α5 ¼ −0:75 is vcr ¼ 0:08 [Fig. 4(f)], which accuratelyseparates unstable [Figs. 5(d) and 5(e)] at v ¼ 0:075 and stabledark solitons at v ¼ 0:085 [Fig. 5(f)]. In particular, the typicalinstability scenarios of the dark solitons is displayed inFigs. 5(d) and 5(e). Under white-noise perturbations, the darksoliton may either transform to a stable one with higher ve-locity [Fig. 5(d); this is slightly different from its counterpartin the local medium, which tends to split into two solitons] orcollapse (not catastrophic) into a broadening dark speck (twocounterpropagating kinks) [Fig. 5(e)].

4. BOUND STATES AND SOLITONCOLLISIONSOne of the important properties of nonlocality is that it sup-ports the formation of bound states of several solitons. Thesimplest bound states of two dark-type solitons for the twoCQ models are displayed in Figs. 6(a)–6(e). Note that thebound states solutions of one dark soliton and one darklikesoliton in CQ model (I) cannot converge in our codes for aseries of trial functions. In all cases, the intensity distributionsexhibit a local maximum (minimum for darklike bright soli-tons) between the two dark-type solitons forming the boundstates. The induced refractive index profile for the stablebound state has the shape of a single waveguide that iscapable of keeping the complex dark-type soliton together[Figs. 6(a), 6(c), and 6(e)]. Multiple intensity oscillations de-velop in profiles of bound states when their velocity ap-proaches the maximal velocity vm. In particular, except forthe bound states of darklike bright solitons in CQ model(I), the maximal velocity vm of the bound states of dark soli-tons is less pronounced than that of single dark solitons,which is in contrast to those in nonlocal Kerr media [38].Nevertheless, the bound states of dark solitons in CQ model(I) imposes nearly the same restrictions for αm5 as those of sin-gle dark solitons [Fig. 2(f)]. At fixed nonlocal lengths σ3;5, thelocal maximum at the center of the bound state grows with α5.It approaches 1 as α5 → αm5 , where the bound state vanishescompletely and the induced waveguide then has the shape of apronounced M-type profile.

For moderate nonlocal media, bound states of two dark so-litons in CQ model (I) can be stable in their entire existenceregions [Fig. 6(f)]. In contrast, the restrictions for bound statesof the darklike bright solitons in CQ model (I) and the darksolitons in CQ model (II) are more strict. Figures 6(b) and6(d) display examples of such unstable bound states. Forthe bound state of the darklike bright solitons in CQ model

−30 0 300.1

0.7

1.3

η

ρ2

−0.4

−0.3

−0.2

−n

−30 0 300.1

0.7

1.3

η

ρ2

−0.4

−0.3

−0.2

−n

0 0.24 0.48 0.720

0.5

1

v

g

σ3,5=0

5

0 0.24 0.48 0.720

8

v

Pr

0

σ3,5=5

0 0.24 0.48 0.72−3

−2

−1

0

v

Mr σ3=15

2.5

5

10

0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

|α5|

v

σ3,5=0 5

Fig. 4. (Color online) Intensity and relative refractive index distributions for the single dark solitons in (a) and (b) in nonlocal CQ model (II).(c) Greyness, (d) renormalized energy flow, and (e) momentum, versus velocity at α5 ¼ −0:75; points marked by squares correspond to (a), andcircles, to (b). (f) Existence (below the squares) and instability regions (below the circles) of dark solitons. In (c)–(f), the blue dashed curvescorrespond to local CQ media (σ3;5 ¼ 0).

Zhou et al. Vol. 28, No. 6 / June 2011 / J. Opt. Soc. Am. B 1587

(I), although the corresponding single soliton is stable at v ¼0:3 > vcr ¼ 0:254 [Fig. 3(f)], a small amount of white-noiseperturbation leads to the catastrophic collapse of the boundstate [Fig. 6(g)]. Also, the perturbed bound state of dark so-litons at v ¼ 0:15 > vcr ¼ 0:08 in CQ model (II) separates intoone stable dark soliton with higher velocity and one broaden-ing dark speck. Further increasing the soliton velocity leads tothe stabilization of these bound states [Figs. 6(h) and 6(j)].

The existence of stable bound states allows us to considerthe collisions of such states, which is impossible in local CQmedia. Figures 7(a)–7(d) display several typical collision sce-narios for the bound states of dark solitons in CQmodel (I). Athigh soliton velocity, the collisions are nearly elastic, i.e., thebound states almost retain their structures and propagationangles after the collision, and acquire only small shifts inthe transverse direction [Fig. 7(a)]. When the bound states

move slower, one can observe inelastic collisions with largecopropagating distances and radiation emitted [Fig. 7(b)].When the velocity approaches 0, the bound states exhibit arepulsion manner [Fig. 7(c)], similar to dark solitons in localmedia. In particular, the collisions become more inelasticwhen α5 → αm5 : it leads to the destruction of both bound states[Fig. 7(d)].

In the case of darklike bright solitons, in contrary to theircounterparts in local media, the collision always leads to thecollapse of the darklike solitons even at maximal velocity vm.Also, collisions between one dark soliton and one darklikebright soliton exhibit the same behavior. Moreover, boundstates of such solitons do not change the picture [Fig. 7(e)].

It is more interesting for the case of dark solitons in CQmodel (II). The collision may lead to the collapse of both so-litons even they are stable and traveling alone. There are

η

ξ

0 75 150

0

150

300

η

ξ

0 35 70

0

150

300

η

ξ

0 50 100

0

200

400

η

ξ

84420

0

350

700

η

ξ

0 30 60

0

330

660

η

ξ

0 39 78

0

450

900

Fig. 5. (Color online) Propagation dynamics of perturbed dark-type solitons in the two nonlocal CQ models with σ3;5 ¼ 5. (a) Dark solitons atv ¼ 0:5 in model (I) with α5 ¼ 0:2; (b),(c) unstable and stable darklike bright solitons at v ¼ 0:25 and v ¼ 0:26 in model (I) with α5 ¼ 0:4, respec-tively; (d),(e) unstable and (f) stable dark solitons with α5 ¼ −0:75. Other parameters are (d),(e) v ¼ 0:075, (f) v ¼ 0:085.

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ρ2

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−n

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ρ2

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300

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220

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ξ

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0

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400

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400

Fig. 6. (Color online) (a)–(e) Intensity and relative refractive index distributions for a bound state of two dark-type solitons in the two nonlocal CQmodels with σ3;5 ¼ 5. (f)–(j) Propagational dynamics of perturbed dark-type solitons depicted in (a)–(e). (a) A dark soliton at v ¼ 0:3 in CQ model(I) with α5 ¼ 0:2; (b),(c) darklike solitons at v ¼ 0:3 and v ¼ 0:33 in CQ model (I) with α5 ¼ 0:4, respectively; (d),(e) dark solitons at v ¼ 0:15 andv ¼ 0:3 in CQ model (II) with α5 ¼ −0:75, respectively.

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richer collision scenarios for bound states. At relatively lowvelocity, the two bound states collapse into one broadeningdark speck with a large number of radiation waves [Fig. 7(f)].As the velocity grows, the collision leads to the existence oftwo weakly localized dark beams [Fig. 7(g)]. At higher veloc-ity, one can see that collapse vanishes and two pairs of fullyseparated beams are formed with intensive radiation emitted[Fig. 7(h)]. In particular, a nearly elastic collision can beobserved as the velocity increases further.

5. CONCLUSIONIn summary, based on the phenomenological model of Miha-lache et al. [26] for nonlocal non-Kerr media, we have studiedthe dark-type solitons, including dark solitons and darklikebright solitons, in two competing nonlocal CQ models: focus-ing cubic and defocusing quintic nonlinearities for CQ model(I) and defocusing cubic and focusing quintic nonlinearitiesfor CQ model (II). Nonlocality of nonlinear response drasti-cally affects the profile, velocity, existence properties, and sta-bility properties of these solitons. Stable bound states of dark-type solitons can exist in the two models. Through numericalsimulations, we reveal that the stability of the single dark-typesolitons strictly follows the stability criterion. Particularly,nonlocality leads to pronounced shrink of existence param-eter region for dark soliton in CQ model (I). Moreover, non-locality exhibits a remarkable destabilizing effect on darklikebright solitons in CQ model (I) and dark solitons in CQ model(II) that have instability regions. We also investigated the col-lisions between the dark-type solitons and their bound states.Our finding enriches the concept of nonlocal dark-type soli-tons, especially for those with instability regions.

ACKNOWLEDGMENTSThe authors gratefully thank the anonymous referees for in-sightful comments and valuable suggestions. This work was

supported by the National Natural Science Foundation ofChina (NSFC) (grant no. 11074059) and the Program of Excel-lence Teams at Harbin Institute of Technology.

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η

ξ

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