Solitons in a medium with lineardissipation and localized gain

Dmitry A. Zezyulin,1,* Yaroslav V. Kartashov,2 and Vladimir V. Konotop1

1Centro de Física Teórica e Computacional, and Departamento de Física, Faculdade de Ciências,Universidade de Lisboa, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal

2ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya,Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

*Corresponding author: zezyulin@cii.fc.ul.pt

Received December 24, 2010; revised January 29, 2011; accepted February 17, 2011;posted February 25, 2011 (Doc. ID 140246); published March 28, 2011

We present a variety of dissipative solitons and breathing modes in a mediumwith localized gain and homogeneouslinear dissipation. The system possesses a number of unusual properties, like exponentially localizedmodes in bothfocusing and defocusing media, existence of modes in focusing media at negative propagation constant values,simultaneous existence of stable symmetric and antisymmetric localized modes when the gain landscape possessestwo local maxima, as well as the existence of stable breathing solutions. © 2011 Optical Society of AmericaOCIS codes: 190.0190, 190.6135.

Dissipative solitons appear in a variety of forms andsettings. They form in laser systems with saturable gainand absorption [1], in homogeneous materials wherelight evolution can be described by the cubic–quinticGinzburg–Landau (GL) equation [2], and in systems withspatially localized gain and uniform nonlinear losses[3–8]. A characteristic feature of all these systems is thatthe existence of dissipative solitons there is mediated bythe competition of gain and losses having different phys-ical origin. Thus, in the cubic–quintic GL equation, soli-tons exist because of exact balance of the linear losses,cubic gain, and quintic losses. Likewise, solitons can alsoform because cubic losses compensate the spatially loca-lized gain. This has been established in one-dimensionalsettings in the presence of a linear localized potential [3]or lattice [4], in two-dimensional problems in planarwaveguide arrays [5] or lattices [6], and even in two-dimensional systems without additional refractive indexmodulations [7].It is therefore important to elucidate simpler physical

settings, where only the linear gain and losses result information of dissipative solitons. This is the goal of thisLetter. More specifically, we report on the existence ofstable dissipative solitons supported by a localized gainin a homogeneous Kerr medium where only the linearlosses are present far from the center of gain landscape.This simple system possesses a number of interestingproperties among which is the existence of localizedmodes even for negative propagation constant values, si-multaneous stability of symmetric and antisymmetricmodes in two-humped gain landscapes, as well as the ex-istence of breathing modes.We consider the nonlinear Schrödinger equation

iqz ¼ −qxx − i½γ0 − γðxÞ�qþ σjqj2q; ð1Þ

where σ ¼ 1 (σ ¼ −1) corresponds to defocusing (focus-ing) nonlinearity, and γðxÞ describes a localized gainconcentrated in the domain (or domains) having a char-acteristic width d. Without loss of generality we rescalethe background dissipation to the value γ0 ¼ 1 and con-sider the situation where max γðxÞ > 1.

We look for spatially localized solutions obeyinglimjxj→∞ jqðz; xÞj ¼ 0 and start with the stationary modesqðz; xÞ ¼ eibzwðxÞ, where wðxÞ ¼ eiθðxÞuðxÞ, with uðxÞ ≥ 0and θðxÞ being real functions. Then, introducing the cur-rent density jðxÞ≡ θxu2 one obtains from (1) the systemdescribing stationary modes

uxx − buþ σu3 −j2

u3 ¼ 0; jx þ ½1 − γðxÞ�u2 ¼ 0: ð2Þ

In order to establish the domain of possible variation ofthe propagation constant b, we now address the limitx → ∞, focusing on the exponentially decaying solutions,i.e., assuming u ∼ Ce−μx, where C and μ are positive con-stants. From the second equation of system (2), it readilyfollows that, in this limit, j ∼ C2e−2μx=2μ. This suggeststhat, in the first equation of system (2), the term σu3

can be neglected in the limit x → ∞, while substitutingthe asymptotic values of u and j in the terms that are left,we obtain b ¼ μ2 − 1=ð4μ2Þ. It follows from this formulathat the propagation constant for localized solutions canbe either positive or negative even if the nonlinearity isfocusing (contrary to what happens in uniform conserva-tive systems or in dissipative systems [3,4,6,7]). Ifb → þ∞, then μ → þ∞, while if b → −∞, then μ → þ0. Re-spectively, there must exist localized modes with μ > 0even at b ¼ 0 while the spatial delocalization occurs inthe limit b → −∞.

The condition of balance between the dissipation andgain (for γ0 ¼ 1) has the form U ¼ R

γðxÞjuj2dx, whereU ¼ R juj2dx is the energy flow. From the balance be-tween the nonlinearity and diffraction, we obtain

bU ¼ −

Zjwxj2dx − σ

Zjwj4dx: ð3Þ

From this formula one readily concludes that, for zeropropagation constant, b ¼ 0, the mode should decay withμ ¼ 2−1=2, i.e., jwxj2≢0, and hence Eq. (3) can be satisfiedonly in the focusing medium (i.e., at σ ¼ −1). Thus, thelinear limit of the problem (i.e., U → 0) corresponds toa localized mode with b ¼ bð0Þ < 0 [indeed, in this case

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R jwj4dx=U ≤ max jwj2 → 0 and can be neglected inEq. (3)]. In the defocusing medium, exponentially loca-lized solutions exist only if b < bð0Þ, while in the focusingmedium dissipative solitons can exist with both positiveand negative propagation constants, i.e., at b > bð0Þ.We start with the case of the gain localized in one spa-

tial domain, setting γðxÞ ¼ ae−x2=d2 (a > 1 characterizes

the gain strength). By fixing the width of the gain land-scape d and changing a, we obtain the dependencies bðaÞand UðaÞ, which are depicted in Fig. 1 for d ¼ 1. Thesedependencies bifurcate from the linear limit correspond-ing to U ¼ 0 and to bð0Þ ≈ −0:61. The very fact ofexistence of localized linear modes with b ¼ bð0Þ in inho-mogeneous gain landscapes is known as the gain-guidingeffect (see, e.g., [9]). In the linear case the guiding occursfor a particular value of a ¼ að0Þ ≈ 1:98. Focusing nonli-nearity diminishes gain at which localization occurs,while in a defocusing medium, a higher gain is requiredfor localization (Fig. 1). In Fig. 1 we also show an exam-ple of a nonlinear mode. Increase of the gain results indecrease of U in the focusing medium that is also accom-panied by expansion of the soliton. In the defocusingmedium the growth of a results in increase of the energyflow and progressive expansion of the soliton out of thegain domain. Solitons are exponentially localized in boththe focusing and defocusing media, as predicted by theanalysis of Eq. (2), which showed that the current resultsin effective renormalization of the propagation constant.Physical understanding of the exponential localization inthe defocusing medium comes also from the observationthat, due to the negative sign in front of j2=u3, the currentresults in the effective self-focusing of the beam.Linear stability analysis (confirmed by the direct

propagation) indicates that the solitons in the focusing

medium are unstable [see Fig. 2(a)], while in the defocus-ing medium they are attractors and can be excited with avariety of inputs ranging from noisy to localized patterns.This difference in the stability can be understoodfrom Eq. (2). The solution represents a flow out of the“source” (i.e., the gain domain). The defocusing nonli-nearity enhances the outflow from the high intensity re-gion, thus contributing to the stability, while the focusingmedium enhances the field concentration in the gaindomain, which stimulates further growth of the peakamplitude.

Next we study the case of the gain landscape withtwo maxima at x ¼ �x0, considering γðxÞ ¼ ae−ðx−x0Þ2=d2þae−ðxþx0Þ2=d2 , where a > 1. Here one also can obtainmodes both for focusing and defocusing media. In Fig. 3we present symmetric (even) and antisymmetric (odd)modes. Both types of solutions bifurcate from the linear

Fig. 1. (Color online) (a) Propagation constant and (b) energyflow versus a for solitons supported by single gain channel.Red and black curves correspond to focusing and defocusingmedia. Dashed line indicates a value corresponding to thelinear limit. Profile of (c) soliton and (d) current density in de-focusing medium at a ¼ 2:5 corresponding to the circles in (a)and (b).

Fig. 2. (Color online) (a) Instability increment δr versus a forsolitons in a single gain channel in focusing medium. Circlesindicate points beyond which solitons exist only in defocusingmedium where they are stable. (b) δrðaÞ for symmetric “s” andantisymmetric “a” solitons in two gain channels with d ¼ 1 indefocusing medium.

Fig. 3. (Color online) (a) Propagation constant and (b) energyflow versus a for solitons supported by the two gain channels atx0 ¼ 6. Solid and dashed lines correspond to the symmetric andantisymmetric modes. Red and black lines correspond to thefocusing and defocusing media. The circles correspond tothe solitons shown in (c), (d). (c) Unstable antisymmetric soli-ton in the focusing medium at a ¼ 1:8. (d) Stable symmetric so-liton in defocusing medium at a ¼ 2:6.

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limit. The values of parameters a and b, at which bifurca-tions take place, are very close [bð0Þs ≈ −0:61, að0Þs ≈ 1:99for symmetric and bð0Þa ≈ −0:60, að0Þa ≈ 1:98 for antisym-metric solitons]. Close to the linear limit and in thefocusing medium the UðaÞ curves for the symmetricand antisymmetric solitons are almost indistinguishable,while in the defocusing medium their energy flows maydiffer considerably [Figs. 3(a) and 3(b)]. The examples ofmodes are shown in Figs. 3(c) and 3(d).Linear stability analysis reveals multiple alternating

stability and instability domains (in a) for both symmetricand antisymmetric modes [Fig. 2(b)]. A remarkable factis that stable symmetric and antisymmetric modes co-exist at the same parameters of the system. Moreover,for the same value of a, both symmetric and anti-symmetric modes in the defocusing medium can beattractors.We found that, for certain values of a, even input

beams evolve into the symmetric stationary mode, whileodd input beams evolve into the antisymmetric mode; seeFigs. 4(a) and 4(b). When one of the modes is unstable, itusually evolves into a stable mode or into pulsatingmode. An example of such behavior is shown in Fig. 4(c),where perturbed unstable antisymmetric mode trans-forms into the breather (we show only the initial stageof the evolution; simulations were performed up toz ¼ 2 · 103). Dynamics shown in Fig. 4(c) is also illu-

strated in Fig. 5, where a corresponding trajectory inthe phase space and dependence UðzÞ is plotted.

Finally, we explore the possibility of switching be-tween stable solutions by changing the control parametera. An example is shown in Fig. 4(d), where the inputbeam corresponds to the stable symmetric mode ata ¼ 2:186. At z ¼ 100 the strength of the gain a isswitched from 2.186 to 2.5. Since the symmetric modeat a ¼ 2:5 is unstable, the switching results in the beamevolution into the stable antisymmetric mode corre-sponding to a ¼ 2:5.

To conclude, we found a variety of unusual propertiesof the solitons described by model (1), such as the exis-tence of exponentially localized modes with negativepropagation constant values in both focusing and de-focusing media, simultaneous stability of symmetric andantisymmetric solitons, and the existence of persistentbreathers.

D. A. Zezyulin and V. V. Konotop were supportedby the grants SFRH/BPD/64835/2009 and PTDC/FIS/112624/2009 (Fundação para a Ciência e a Tecnologia,Portugal).

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Fig. 5. The (a) trajectory and (b) dependence UðzÞ corre-sponding to the dynamics shown in Fig. 4(c). The bold lineshows the limit cycle.

Fig. 4. Excitation of stable (a) symmetric and (b) antisym-metric solitons by symmetric and antisymmetric input beamsin a system with two gain channels for a ¼ 2:391. (c) Excitationof a breather at a ¼ 2:61 starting with the unstable antisym-metric soliton. (d) Switching between stable symmetric andantisymmetric solitons stimulated by abrupt change of a atz ¼ 100.

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