Spatial solitons supported by localized gain [Invited]

Spatial solitons supported by localized gain [Invited]

Boris A. Malomed

Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University,Tel Aviv 69978, Israel ([email protected])

Received June 2, 2014; accepted July 28, 2014;posted August 8, 2014 (Doc. ID 213280); published September 25, 2014

The creation of stable 1D and 2D localizedmodes in lossy nonlinear media is a fundamental problem in optics andplasmonics. This article gives a mini review of theoretical methods elaborated on for this purpose, using localizedgain applied at one or several hot spots (HS). The introduction surveys a broad class of models for which thisapproach was developed. Other sections focus in some detail on basic 1D continuous and discrete systems, wherethe results can be obtained, partly or fully, in an analytical form (and verified by comparison with numericalresults), which provides deeper insight into the nonlinear dynamics of optical beams in dissipative nonlinearmedia. Considered, in particular, are the single and double HS in the usual waveguide with the self-focusing(SF) or self-defocusing (SDF) Kerr nonlinearity, which gives rise to rather sophisticated results in spite of apparentsimplicity of the model, solitons attached to a PT -symmetric dipole embedded into the SF or SDF medium, gapsolitons pinned to an HS in a Bragg grating, and discrete solitons in a 1D lattice with a hot site. © 2014 OpticalSociety of America

OCIS codes: (190.6135) Spatial solitons; (240.6680) Surface plasmons; (230.4480) Optical amplifiers;(190.4360) Nonlinear optics, devices.http://dx.doi.org/10.1364/JOSAB.31.002460

1. INTRODUCTION AND BASIC MODELSSpatial dissipative solitons (SDS) are self-trapped beams oflight [1–4] or plasmonic waves [5–13] propagating in planaror bulk waveguides. Recently, they also have been predictedand observed as condensed exciton–polariton states in semi-conductor microcavities [14–18], and in plasmonic nanostruc-tures [19,20]. They result from the balance between diffractionand self-focusing (SF) nonlinearity, which is maintained si-multaneously with the balance between the material lossand compensating gain. Due to their basic nature, SDS aremodes of profound significance to nonlinear photonics (opticsand plasmonics), as concerns the fundamental studies andpotential applications alike. In particular, a straightforwardpossibility is to use sufficiently narrow SDS beams as signalcarriers in all-optical data-processing schemes. This applica-tion, as well as other settings in which the solitons occur,stresses the importance of stabilization of the SDS modesand of the development of enabling techniques for the gener-ation and steering of such beams.

In terms of the theoretical description, basic models of theSDS dynamics make use of complex Ginzburg–Landau equa-tions (CGLE). The prototypical one is the CGLE with cubicnonlinearity, which includes the conservative paraxial-diffrac-tion and Kerr terms, nonlinear (cubic) loss with coefficientϵ > 0, which represents two-photon absorption in themedium, and the spatially uniform linear gain, with strengthγ > 0, which aims to compensate the loss [1,2]:

∂u∂z

� i2∇2

⊥u − �ϵ − iβ�juj2u� γu: (1)

Here, u is the complex amplitude of the electromagnetic wavein the spatial domain, and z is the propagation distance;the paraxial-diffraction operator ∇2

⊥ acts on transverse

coordinates �x; y� in the case of the propagation in the bulk,or on the single coordinate, x, in the planar waveguide. Ac-cordingly, Eq. (1) is considered as a 2D or 1D equation in thosetwo cases. The equation is normalized so that the diffractioncoefficient is 1, while β is the Kerr coefficient, β > 0 and β < 0corresponding to the SF and self-defocusing (SDF) signs ofthe nonlinearity, respectively.

A more general version of the CGLE may include an imagi-nary part of the diffraction coefficient [21–23], which is essen-tial, in particular, for the use of the CGLE as a model of thetraveling-wave convection [24,25]. However, in optical modelsthat coefficient, which would represent diffusivity of photons,is usually absent.

A well-known fact is that the 1D version of Eq. (1) gives riseto an exact solution in the form of an exact chirped SDS,which is often called a Pereira–Stenflo soliton [26,27]:

u�x; z� � Aeikz�sech�κx��1�iμ; (2)

A2 � 3γ∕�2ϵ�; κ2 � γ∕μ; k � �γ∕2��μ−1 − μ�; (3)

where the chirp coefficient is

μ ��3β∕2ϵ�2 � 2

q− 3β∕�2ϵ�: (4)

This exact solution is subject to obvious instability, due to theaction of the uniform linear gain on the zero background farfrom the soliton’s core. Therefore, an important problem is thedesign of physically relevant models, which may producestable SDS.

One possibility is to achieve full stabilization of the solitonsin systems of linearly coupled CGLEs modeling dual-corewaveguides, with the linear gain and loss acting in different

2460 J. Opt. Soc. Am. B / Vol. 31, No. 10 / October 2014 B. A. Malomed

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http://dx.doi.org/10.1364/JOSAB.31.002460

cores [12, 28–34]. This includes, inter alia, a PT -symmetricversion of the system that features the exact balance betweenthe gain and loss [35,36]. The simplest example of such a sta-bilization mechanism is offered by the following coupledCGLE system [30]:

∂u∂z

� i2∇2

⊥u − �ϵ − iβ�juj2u� γu� iλv; (5)

∂v∂z

� �iq − Γ�v� iλu; (6)

where λ is the linear-coupling coefficient, v�x; z� and Γ > 0 arethe electromagnetic-wave amplitude and the linear-loss rate,respectively, in the stabilizing dissipative core, and q is a pos-sible wavenumber mismatch between the cores. In the case ofq � 0, the zero background is stable in the framework ofEqs. (5) and (6) under condition

γ < Γ < λ2∕γ: (7)

The same ansatz in Eq. (2), which produced the Pereira–Stenflo soliton of the single CGLE, yields an exact solutionof the coupled system in Eqs. (5) and (6):

fu�x; z�; v�x; z�g � fA;Bgeikz�sech�κx��1�iμ; (8)

with chirp μ given by the same expression in Eq. (4) as above,and

B � iλ�Γ� i�k − q��−1A: (9)

A stable soliton is obtained if a pair of distinct solutions arefound, compatible with the condition of the stability for thezero background [which is Eq. (7) in the case of q � 0], in-stead of the single solution in the case of Eq. (3). Then thesoliton with the larger amplitude is stable, coexisting, as anattractor, with the stable zero solution, while the additionalsoliton with a smaller amplitude plays the role of an unstableseparatrix, which delineates the boundary between attractionbasins of the two coexisting stable solutions [30].

In the case of q � 0, the aforementioned condition of theexistence of two solutions reduces to

γΓ�1 − μ2� > 4μ2��λ2 − γΓ� � 2Γ�Γ − γ��: (10)

In particular, it follows from Eqs. (10) and (4) that a relatednecessary condition, μ < 1, implies ϵ < 3β, i.e., the Kerr coef-ficient, β, must feature the SF sign, and the cubic loss coeffi-cient, ϵ, must be sufficiently small in comparison with β. Ifinequality μ < 1 holds, and zero background is close to itsstability boundaries, i.e., 0 < Γ − γ ≪ γ and 0 < λ2 − γΓ ≪ γ2

[see Eq. (7)], parameters of the stable soliton with the largeramplitude are

κst ≈��γ

μ

r1 − μ2

1� μ2; Ast ≈

��3γ2ϵ

r1 − μ2

1� μ2; Bst ≈

2μ

�1 − iμ�2 A;

while propagation constant k is given, in the first approxima-tion, by Eq. (3). In the same case, the unstable separatrix sol-iton has a small amplitude and large width, while its chirpkeeps the same value as in Eq. (4):

κ2sep ≈2�Γ − γ�γ�1 − μ2�

��λ2 − γΓ

q; ksep ≈

��λ2 − γΓ

q;

A2sep ≈

3μ2ϵ

κ2sep; Bsep ≈ iAsep:

Getting back to models based on the single CGLE, stablesolitons also can be generated by the equation with cubic gain“sandwiched” between linear and quintic loss terms, whichcorresponds to the following generalization of Eq. (1):

∂u∂z

� i2∇2

⊥u� �ϵ3 � iβ3�juj2u − �ϵ5 � iβ5�juj4u − Γu; (11)

with ϵ3 > 0, ϵ5 > 0, Γ > 0, and β5 ≥ 0. The linear loss, repre-sented by coefficient Γ, provides for the stability of the zerosolution to Eq. (11). This cubic-quintic (CQ) CGLE was firstproposed, in a phenomenological (actually, 2D) form, byPetviashvili and Sergeev [37]. Later, it was demonstrated thatthe CQ model may be realized in optics as a combination oflinear amplification and saturable absorption [38–40]. Stabledissipative solitons supported by this model were investigatedin detail by means of numerical and analytical methods[41–46].

The subject of the present mini review is the developmentof another method for creating stable localized modes, whichmakes use of linear gain applied at a hot spot (HS), i.e., a local-ized amplifying region embedded into a bulk lossy waveguide.The experimental technique, which allows one to create local-ized gain by means of strongly inhomogeneous distributionsof dopants implanted into the lossy waveguide and producesthe gain if pumped by an external source of light, is wellknown [47]. Another possibility is even more feasible and ver-satile: the dopant density may be uniform, while the externalpump beam is focused on the location where the HS should becreated. Furthermore, the latter technique makes it possibleto create effectively moving HS, if the pump beam is appliedin an oblique direction, in terms of the spatial-domaindescription.

Supporting dissipative solitons by the localized gain wasfirst proposed not in the framework of CGLEs, but for agap soliton pinned to an HS in a lossy Bragg grating (BG)[48]. In terms of the spatial-domain dynamics, the respectivemodel is based on the system of coupled-mode equations(CMEs) for counterpropagating waves, u�x; z� and u�x; z�,coupled by the Bragg reflection:

iuz � iux � v� �juj2 � 2jvj2�u � −iγu� i�Γ1 � iΓ2�δ�x�u;(12)

ivz − ivx � u� �jvj2 � 2juj2�v � −iγv� i�Γ1 � iΓ2�δ�x�v;(13)

where the tilt of the light beam and the reflection coefficientsare normalized to be 1, the nonlinear terms account for theself- and cross-phase modulation induced by the Kerr effect,γ > 0 is the linear-loss parameter, Γ1 > 0 represents the localgain applied at the HS [x � 0, δ�x� being the Dirac’s delta func-tion], and the imaginary part of the gain coefficient, Γ2 ≥ 0,accounts for a possible attractive potential induced by theHS (it approximates a local increase of the refractive indexaround the HS).

B. A. Malomed Vol. 31, No. 10 / October 2014 / J. Opt. Soc. Am. B 2461

As mentioned in [48] too, and for the first time investigatedin detail in [49], the HS embedded into the usual planar wave-guide is described by the following modification of Eq. (1):

∂u∂z

� i2∇2

⊥u − �ϵ − iβ�juj2u − γu� �Γ1 � iΓ2�δ�x�u; (14)

where, as well as in Eqs. (12) and (13), Γ1 > 0 is assumed, andthe negative sign in front of γ ≥ 0 represents the linear loss inthe bulk waveguide. Another HS model, based on the 1DCGLE with the CQ nonlinearity, was introduced in [50]:

∂u∂z

� i2∂2u∂x2

� ijuj2u − iβ5juj2u − γu� Γe−x2∕w2 juj2u; (15)

where β5 > 0 represents the quintic self-defocusing term,γ > 0 and Γ > 0 are, as above, strengths of the bulk lossesand localized cubic gain, and w is the width of the HS (anapproximation corresponding to w → 0, with the HS in theform of the delta function, may be applied here too). Whilesolitons in uniform media, supported by the cubic gain, arealways unstable against the blowup in the absence of the quin-tic loss [51], the analysis reported in [50] demonstrates that,counterintuitively, stable dissipative localized modes in theuniform lossy medium may be supported by the unsaturated

localized cubic gain in the model based on Eq. (15).In addition to the direct linear gain assumed in the above-

mentioned models, losses in photonic media may be compen-sated by parametric amplification, which, unlike the directgain, is sensitive to the phase of the signal [52,53]. This mecha-nism can be used for the creation of an HS, if the parametricgain is applied in a narrow segment of the waveguide. As pro-posed in [54], the respective 1D model is based on the follow-ing equation, cf. Eqs. (14) and (15):

∂u∂z

� i2∂2u∂x2

� ijuj2u − �ϵ − iβ�juj2u − �γ − iq�u� Γe−x2∕w2u�;

(16)

where u� is the complex conjugate field, q is a real phase-mismatch parameter, and, as well as in Eq. (15), the HSmay be approximated by the delta function in the limitof w → 0.

Models combining the localized gain and the uniformly dis-tributed Kerr nonlinearity and linear loss have been recentlydeveloped in various directions. In particular, 1D models withtwo or multiple HS [55–59] and periodic amplifying structures[60,61], as well as extended patterns [62,63], have been stud-ied, chiefly by means of numerical methods. The numericalanalysis also has made it possible to study 2D settings inwhich, most notably, stable localized vortices are supportedby the gain confined to an annular-shaped area [64–68].The parametric amplification applied at a ring may supportstable vortices, too, provided that the pump 2D beam itselfhas an inner vortical structure [68].

A similar model has been elaborated for exciton–polaritoncondensates in microcavities, which is based on coupled evo-lution equations for the exciton �ϕX � and cavity-photon �ϕc�wave functions [15]:

i∂ϕX

∂t� −ϵϕX − iγXϕX � Ωϕc � jϕX j2ϕX ;

i∂ϕc

∂t� −

12∂2ϕc

∂x2− iγcϕc � ΩϕX � f �x�:

Here ϵ is the difference in the chemical potentials between thetwo wave functions, Ω is the coefficient of the Rabi couplingbetween them, γX;c are the respective dissipative constants,terms jϕX j2ϕX and −�1∕2�∂2ϕc∕∂x2 represent, respectively,the repulsion between excitons and diffraction of photons,and f �x� is a localized photonic pump. Various SDS-like modespinned to a narrow pump [which, in principle may be approxi-mated by δ�x�], were found in [15].

Another ramification of the topic is the development ofsymmetric combinations of hot and cold spots, which offera realization of the concept of PT -symmetric systems in op-tical media and were proposed and built as settings integratingthe balanced spatially separated gain and loss with a spatiallysymmetric profile of the local refractive index [69–72]. Thestudy of solitons in nonlinear PT -symmetric settings hasdrawn a great deal of attention [35,36,73–78]. In particular,it is possible to consider the 1D model in the form of a sym-metric pair of hot and cold spots described by two delta func-tions embedded into a bulk conservative medium with cubicnonlinearity [79]. A limit case of this setting, which admitsexact analytical solutions for PT -symmetric solitons, corre-sponds to a PT dipole, which is represented by the derivativeof the delta function in the following 1D equation [80]:

i∂u∂z

� −

12∂2u∂x2

− σjuj2u − ε0uδ�x� � iγuδ0�x�: (17)

Here σ � �1 and −1 correspond to the SF and SDF bulk non-linearity, respectively, ε0 ≥ 0 is the strength of the attractivepotential, which is a natural conservative component of thePT dipole, and γ is the strength of the PT dipole.

It is also natural to consider discrete photonic settings(lattices), which appear, in the form of discrete CGLEs, asmodels of arrayed optical [81–87] or plasmonics [88,89] wave-guides. In this context, lattice counterparts of the HS amountto a single [90] or several [91] amplified site(s) embedded intoa 1D or 2D [92] lossy array. Being interested in tightly local-ized discrete states, one can additionally simplify the model byassuming that the nonlinearity is carried only by the activecores, which gives rise to the following version of the discreteCGLE, written here in the general 2D form [92]:

dum;n

dz� i

2�um−1;n � um�1;n � um;n−1 � um;n�1 − 4um;n�;

− γum;n � ��Γ1 � iΓ2� � �iB − E�jum;nj2�δm;0δn;0um;n;

(18)

where m;n � 0;1;2;… are discrete coordinates on thelattice, δm;0 and δn;0 are the Kronecker’s symbols, and the co-efficient of the linear coupling between adjacent cores isscaled to be 1. As above, γ > 0 is the linear loss in the bulklattice, Γ1 > 0 and Γ2 ≥ 0 represent the linear gain and linearpotential applied at the HS site (m � n � 0), while B and Eaccount for the SF (B > 0) or SDF (B < 0) Kerr nonlinearityand nonlinear loss (E > 0) or gain (E < 0) acting at the HS


(the unsaturated cubic gain may be a meaningful feature inthis setting [92]).

The PT symmetry can be introduced, too, in the frameworkof the lattice system. In particular, a discrete counterpart ofthe 1D continuous model in Eq. (17) with the PT -symmetricdipole was recently elaborated in [93]:

idun

dz� −�Cn;n−1un−1 � Cn�1;nun�1� − gnjunj2un � iκnun;

(19)

where the PT dimer (discrete dipole) embedded into theHamiltonian lattice is represented by κn � �κ at n � 0, −κat n � 1, and 0 at n ≠ 0; 1. A counterpart of the delta func-tional attractive potential in Eq. (17) corresponds to a localdefect in the intersite couplings: C1;0 � Cd, Cn;n−1 � C0 ≠Cd at n ≠ 1. Last, the nonlinearity is assumed to be carriedsolely by the dimer embedded into the lattice: gn � g atn � 0; 1, and 0 at n ≠ 0; 1, cf. Eq. (18). Equation (19) admitsexact analytical solutions for all PT -symmetric and -antisym-metric discrete solitons pinned to the dimer.

In addition to the HS, one can naturally define a “warmspot” (WS) in the 2D CGLE with the CQ nonlinearity, wherethe coefficient of the linear loss is spatially profiled with aminimum at the WS (r � 0) [64]. Thus, the equation may betaken as the 2D version of Eq. (11) with

Γ�r� � Γ0 � Γ2r2; (20)

where r is the radial coordinate, coefficients Γ0 and Γ2 beingpositive. This seemingly simple model gives rise to a greatvariety of stable 2D modes pinned to the WS. Dependingon values of parameters in Eqs. (11) and (20), these maybe simple vortices, rotating elliptic, eccentric, and slantedvortices, spinning crescents, etc. [64].

The use of the spatial modulation of loss coefficients opensanother way for the stabilization of the SDS: as shown in [94],the solitons may be readily made stable if the spatially

uniform linear gain is combined with the local strength ofthe cubic loss, ϵ�r�, growing from the center to periphery atany rate faster than rD, where r is the distance from the centerand D the spatial dimension. This setting is described by thefollowing modification of Eq. (1):

∂u∂z

� i2∇2

⊥u − �ϵ�r� − iβ�juj2u� γu; (21)

where, as said above, γ and ϵ�r� are positive, so thatlimr→∞�rD∕ϵ�r�� 0, for D � 1 or 2.

The models outlined above feature either the pure cubic orCQ nonlinearities. It also is natural to consider second-harmonic (SH) generating systems with the quadratic (χ�2�)nonlinearity, in which linear losses (γ1 and γ2), acting on boththe fundamental frequency (FF) and SH components (q1 andq2, respectively), may be compensated by localized gain ap-plied at the HS. If the gain acts on the FF wave, the situationis qualitatively similar to that produced by Eq. (14); in thiscase, the χ�2� system may be approximately reduced to theusual cubic one by means of the cascading approximation,which assumes a large mismatch (β) between the FF and SH.A novel setting corresponds to the system in which the local-ized gain acts on the SH component. This model was recentlyintroduced in [95]:

i∂q1∂z

� −

12∂2q1∂x2

− q�1q2 − iγ1q1;

i∂q2∂z

� −

14∂2q2∂x2

� βq2 − q21 − iγ1q1 � iΓ�x�q2;

where q�1 is the complex conjugate field, and Γ�x� is the profileof the HS.

This mini review aims to present a survey of basic resultsobtained for SDS pinned to HS in the class of models outlinedabove. In view of the limited length of this article, stress isplaced on the most fundamental results that can be obtainedin an analytical or semi-analytical form, in combination withthe related numerical findings, thus providing deeper insightinto the dynamics of the underlying photonic systems. Thepossibility of obtaining many essential results in an analyticalform is an asset of these models. First, in Section 2, findingsare summarized for the most fundamental 1D model based onEq. (14), which is followed, in Section 3, by the considerationof the PT -symmetric system in Eq. (17). The BG model inEqs. (12) and (13) is the subject of Section 4. Section 5 dealswith the 1D version of the discrete model in Eq. (18). Thearticle is concluded by Section 6.

2. DISSIPATIVE SOLITONS PINNED TO HOTSPOTS IN ORDINARY WAVEGUIDEThe presentation in this section is focused on the basic modelin Eq. (14) and its extension for two HS, chiefly followingworks [49] and [55], for the settings with a single and doubleHS, respectively. Analytical and numerical results are pre-sented, which highlight the most fundamental properties ofSDS supported by the tightly localized gain embedded intolossy optical media.

A. Analytical Considerations

1. Exact ResultsStationary solutions to Eq. (14) are looked for as u�x; t� �eikzU�x�, where complex function U�x� satisfies an ordinarydifferential equation:

�γ � ik�U � i2d2Udx2

− �ϵ − iβ�jUj2U; (22)

at x ≠ 0, supplemented by the boundary condition (b.c.) atx � 0, which is generated by the integration of Eq. (14) inan infinitesimal vicinity of x � 0:

limx→�0

ddx

U�x� � �iΓ1 − Γ2�U�x � 0�; (23)

assuming even stationary solutions, U�−x� � U�x�.As seen from the expression for A2 in Eq. (3) [recall that

Eqs. (1) and (14) have opposite signs in front of γ],Eq. (22) with γ > 0 and ϵ > 0 cannot be solved by a sechansatz similar to that in Eq. (2). As an alternative, sech canbe replaced by 1∕ sinh:

U�x� � A�sinh�κ�jxj � ξ��−�1�iμ�; (24)

where offset ξ > 0 prevents the existence of the singularity.This ansatz yields an exact codimension-one solution toEq. (22) with b.c. in Eq. (23), i.e., a solution that is valid undera special constraint imposed on coefficients of the system:


Γ1∕Γ2 − 2Γ2∕Γ1 � 3β∕ϵ: (25)

Parameters of this solution are

A2 � 3γ2ϵ

; κ2 � γΓ2

Γ1; μ � −

Γ1

Γ2; k � γ

2

�Γ2

Γ1−

Γ1

Γ2

�;

(26)

ξ � 12

��Γ1

γΓ2

sln� ��

Γ1Γ2p � ��

γp��

Γ1Γ2p

−

��γ

p�: (27)

The squared amplitude of the solution is

jU�x � 0�j2 � �3∕2ϵ��Γ1Γ2 − γ�: (28)

The main characteristic of the localized beam is its totalpower:

P �Z �∞

−∞ju�x�j2dx: (29)

For the solution given by Eqs. (24)–(27),

P � �3∕ϵ��Γ1

p � ��Γ1

p−

��γ∕Γ2

p �: (30)

Obviously, solution (24) exists if it yields jU�x � 0�j2 > 0 andP > 0, i.e.,

Γ1 > �Γ1�thr ≡ γ∕Γ2: (31)

The meaning of the threshold condition in Eq. (31) is that, tosupport the stable pinned soliton, the localized gain (Γ1) mustbe sufficiently large in comparison with the background loss,γ. It is relevant to mention that, according to Eq. (25), theexact solution given by Eqs. (24), (26), and (27) emerges atthe threshold (31) in the SDF medium, with β < 0, providedthat condition γ <

��2

pΓ22 holds. In the opposite case,

γ >��2

pΓ22, the threshold is realized in the SF medium,

with β > 0.

2. Exact Results for γ = 0 (No Linear Background Loss)The above analytical solution admits a nontrivial limit forγ → 0, which implies that the local gain compensates onlythe nonlinear loss, accounted for by term ∼ϵ in Eq. (14). Inthis limit, the pinned state is weakly localized, instead ofthe exponentially localized one in Eq. (24):

Uγ�0�x� ��32ϵ

r ��Γ1∕Γ2

p�jxj � Γ−1

2 �1�iμ ; k � 0; (32)

with μ given by Eq. (26) (an overall phase shift is droppedhere). Note that the existence of solution (32) does not requireany threshold condition, unlike Eq. (31). This weakly localizedstate is a physically meaningful one, as its total power (29)converges, P�γ � 0� � 3Γ1∕ϵ. Note that this power does notdepend on the local potential strength, Γ2, unlike the genericexpression (30).

3. Perturbative Results for the Self-Defocusing MediumIn the limit case when the loss and gain vanish, γ � ϵ �Γ1 � 0, Eq. (24) goes over into an exact one for a localizedmode pinned by the attractive potential in the self-defocusingmedium (SDF) (with β < 0):

U�x� ��2k∕jβj

psinh

� ��2k

p�jxj � ξ0�

� ; (33)

ξ0 �1

2��2k

p ln�Γ2 �

��2k

p

Γ2 −��2k

p�; (34)

jU�x � 0�j2 � jβj−1�Γ2 − 2k�; (35)

which exists in interval 0 < k < �1∕2�Γ22 of the propagation

constant. The total power of this solution is

P0 � �2∕jβj��Γ2 −

��2k

p �: (36)

In the limit of k � 0, when amplitude in Eq. (35) and power inEq. (36) attain their maxima, the solution degenerates fromthe exponentially localized one into a weakly localized state[cf. Eq. (32)],

Uk�0�x� �1��

jβjp

�jxj � Γ−12 �

; (37)

whose total power converges, P0�k � 0� � 2Γ2∕jβj, asper Eq. (36).

The exact solutions given by Eqs. (33)–(37), which aregeneric ones in the conservative model [no spacial constraint,such as in Eq. (25), is required], may be used to construct anapproximate solution to the full system of Eqs. (22) and (23),assuming that the gain and loss parameters (Γ1, γ, and ϵ) areall small. To this end, one can use the balance equation for thetotal power:

dPdz

� −2γP − 2ϵZ �∞

−∞ju�x�j4dx� 2Γ1ju�x � 0�j2 � 0: (38)

The substitution, in the zero-order approximation, of solu-tion (33), (34) into Eq. (38) yields the gain strength, whichis required to compensate the linear and nonlinear losses inthe solution with propagation constant k:

Γ1 �2γ

Γ2 ��2k

p � 2ϵ3jβj

�Γ2 −

��2k

p ��Γ2 � 2

��2k

p �Γ2 �

��2k

p : (39)

As follows from Eq. (39), with the decrease of k from the larg-est possible value, �1∕2�Γ2

2, to 0, the necessary gain increasesfrom the minimum, which exactly coincides with the thresh-old in Eq. (31), to the largest value at which the perturbativetreatment admits the existence of the stationary pinned mode:

�Γ1�max � 2γ∕Γ2 � 2ϵΓ2∕�3jβj�: (40)

The respective total power grows from 0 to the above-mentioned maximum, 2Γ2∕jβj.


It is expected that, at Γ1 exceeding the limit value inEq. (40) admitted by the stationary mode, the solution be-comes nonstationary, with the pinned mode emitting radiationwaves, which makes the effective loss larger and thus balan-ces Γ1 − �Γ1�max. However, this issue was not studied in detail.

The perturbative result clearly suggests that, in the lossySDF medium with the local gain, the pinned modes existnot only under the special condition (25), at which they areavailable in the exact form, but as fully generic solutionstoo. Furthermore, the increase of the power with the gainstrength implies that the modes are, most plausibly,stable ones.

4. Perturbative Results for the Self-Focusing MediumIn the case of β > 0, which corresponds to the SF sign of thecubic nonlinearity, a commonly known exact solution forthe pinned mode in the absence of the loss and gain,γ � ϵ � Γ1 � 0, is

U�x� ��2k∕jβj

psech

� ��2k

p�jxj � ξ0�

�; (41)

ξ0 �1

2��2k

p ln� ��

2kp

� Γ2��2k

p− Γ2

�; (42)

jU�x � 0�j2 � jβj−1�2k − Γ22�; (43)

with the total power:

P0 � �2∕β��

2kp

− Γ2

�; (44)

which exists for propagation constants k > �1∕2�Γ22, cf.

Eqs. (33)–(36). In this case, the power-balance condition inEq. (38) yields a result that is essentially different from itscounterpart (39):

Γ1 �2γ��

2kp

� Γ2

� 2ϵ3β

� ��2k

p− Γ2

��2

��2k

p� Γ2

��2k

p� Γ2

: (45)

Straightforward consideration of Eq. (45) reveals the differ-ence of the situation from that considered above for theSDF medium: if the strength of the nonlinear loss is relativelysmall,

ϵ < ϵcr � �β∕2Γ22�γ; (46)

the growth of power in Eq. (44) from zero at��2k

p� Γ2 with

the increase of k initially (at small values of P0) requiresnot the increase of the gain strength from the threshold value(31) to Γ1 > �Γ1�thr, but, on the contrary, decrease of Γ1 toΓ1 < �Γ1�thr [48]. Only at ϵ > ϵcr [see Eq. (46)] the power growswith Γ1 starting from Γ1 � �Γ1�thr.

5. Stability of Zero Solution, and Its Relation to theExistence of Pinned SolitonsIt is possible to check the stability of the zero solution, whichis closely related to the soliton’s stability, as noted above. Tothis end, one should use the linearized version of Eq. (14):

∂ulin

∂z� i

2∂2ulin

∂x2− γulin � �Γ1 � iΓ2�δ�x�ulin: (47)

The critical role is played by localized eigenmodes of Eq. (47):

ulin�x; t� � u0eΛzeiljxj−λjxj; (48)

where u0 is an arbitrary amplitude, localization parameter λmust be positive, l is a wavenumber, and Λ is a complex in-stability growth rate. Straightforward analysis yields [49]

λ − il � Γ2 − iΓ1; Λ � �i∕2��Γ2 − iΓ1�2 − γ: (49)

It follows from Eq. (49) that the stability condition for the zerosolution, Re�Λ� < 0, amounts to inequality γ > Γ1Γ2, which isexactly opposite to Eq. (31). In fact, Eq. (28) demonstratesthat the exact pinned-soliton solution given by Eqs. (24)–(27) emerges via the standard forward (alias supercritical)pitchfork bifurcation [96], precisely at the point where thezero solution loses its stability to the local perturbation.The analysis of Eq. (39) demonstrates that the same happenswith the perturbative solution (33) and (34) in the SDF model.On the contrary, the analysis of Eq. (45) has revealed abovethat the same transition happens to the perturbative solu-tion (41) and (42) in the SF medium only at ϵ > ϵcr [seeEq. (46)], while at ϵ > ϵcr the pitchfork bifurcation is of thebackward (alias subcritical [96]) type, featuring the powerthat originally grows with the decrease of the gain strength.Accordingly, the pinned modes emerging from the subcriticalbifurcation are unstable. However, the contribution of thenonlinear loss (ϵ > 0) eventually leads to the turn of the sol-ution branch forward and its stabilization at the turning point.For very small ϵ, the turning point determined by Eq. (45) islocated at Γ1 ≈ 4

��2γϵ∕�3β�

pand P0 ≈

��6γ∕�βϵ�

p.

Last, note that the zero solution is never destabilized, andthe stable pinned soliton does not emerge, in the absence ofthe local attractive potential, i.e., at Γ2 ≤ 0.

B. Numerical Results

1. Self-Trapping and Stability of Pinned SolitonsNumerical analysis of the model based on Eq. (14) was per-formed with the delta function replaced by its Gaussianapproximation:

~δ�x� � � ��π

pσ�−1 exp�−x2∕σ2�; (50)

with finite width σ. The shape of a typical analytical solu-tion (24)–(28) for the pinned soliton, and a set of approxima-tions to it provided by the use of approximation (50), aredisplayed in Fig. 1. All these pinned states are stable, aswas checked by simulations of their perturbed evolution inthe framework of Eq. (14) with δ�x� replaced by ~δ�x�.

The minimum (threshold) value of the local gain strength,�Γ1�thr, which is necessary for the existence of stable pinnedsolitons is an important characteristic of the setting [seeEq. (31)]. Figure 2 displays the dependence of �Γ1�thr onthe strength of the background loss, characterized by

��γ

p,

for β � 0 and three fixed values of the local potential’sstrength, Γ2 � �1; 0;−1 (in fact, the solutions correspondingto Γ2 � −1 are unstable, as they are repelled by the HS). Inaddition, �Γ1�thr is also shown as a function of

��γ

punder


constraint (25), which amounts to Γ2 � Γ1∕��2

p, in the case of

β � 0. The corresponding analytical prediction, as given byEq. (31), is virtually identical to its numerical counterpart,despite the difference of the approximation in Eq. (50) fromthe ideal delta function.

Figure 2 corroborates that, as said above, the analytical sol-utions represent only a particular case of the family of genericdissipative solitons that can be found in the numerical form. Inparticular, the solution produces narrow and tall stable pinnedsolitons at large values of Γ1. All the pinned solitons, includingweakly localized ones predicted by the analytical solution (32)for γ � 0, are stable at Γ1 > �Γ1�thr and Γ2 > 0.

The situation is essentially different for large σ in Eq. (50),i.e., when the local gain is supplied in a broad region. In that

case, simulations do not demonstrate self-trapping into sta-tionary solitons; instead, a generic outcome is the formationof stable breathers featuring regular intrinsic oscillations, thebreather’s width being on the same order of magnitude as σ(see a typical example in Fig. 3). It seems plausible that, withthe increase of σ, the static pinned soliton is destabilized viathe Hopf bifurcation [96], which gives rise to the stablebreather.

C. The Model with Double Hot SpotThe extension of Eq. (14) for two mutually symmetric HSsseparated by distance 2L was introduced in [55]:

∂u∂z

� i2∂2u∂x2

− γu − �ϵ − iβ�juj2u

� �Γ1 � iΓ2��δ�x� L� � δ�x − L��u: (51)

Numerical analysis has demonstrated that stationary symmet-ric solutions of this equation (they are not available in an ana-lytical form) are stable (see a typical example in Fig. 4), whileall antisymmetric states are unstable, spontaneously trans-forming into their symmetric counterparts.

−20 −15 −10 −5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

x

|u|

sinh Ansatzσ=0.06σ=0.08σ=0.10σ=0.12σ=0.14

Fig. 1. Exact solution for the pinned soliton given by Eqs. (24)–(28),and a set of approximations generated by the regularized delta func-tion defined as per Eq. (50). All the profiles represent stable solutions.Other parameters are β � 0, γ � 0.25, Γ1 � 0.6155, and Γ2 � Γ1∕

��2

p,

as per Eq. (25).

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

γ1/2

Γ 1

Γ2= −1

Γ2= 0

Γ2= Γ

1/21/2

Γ2= 1

Fig. 2. Chains of symbols show the minimum value of the local gain,Γ1, which is necessary for the creation of stationary pinned solitons inthe framework of Eq. (14), as a function of the background loss (γ),with β � 0, ϵ � 1, and δ�x� approximated as per Eq. (50) with σ � 0.1.The strength of the local potential, Γ2, is fixed as indicated in the box.For Γ2 � 1 and 0, the lines are guides for the eye, while the straightline for the case of Γ2 � Γ1∕

��2

p, which corresponds to Eq. (25) with

β � 0, is the analytical prediction given by Eq. (31).

Fig. 3. Typical example of a robust breather produced by simula-tions of Eq. (14) with δ�x� replaced by approximation ~δ�x� as perEq. (50) with σ � 2, the other parameters being β � ϵ � Γ2 � 1,Γ1 � 4, and γ � 0.1.

−30 −20 −10 0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

x

|u|

Fig. 4. A stable symmetric mode generated by Eq. (51) with the deltafunctions approximated as per Eq. (50) with σ � 0.1. Other parame-ters are β � 0, ϵ � 1, γ � 0.057, Γ1 � 0.334, Γ2 � 0.236, and L � 8.


As shown above in Fig. 3, Eq. (14) with the single HS de-scribed by Eq. (50), where σ is large enough, supports breath-ers, instead of stationary pinned modes. Simulations ofEq. (51) demonstrate that a pair of such broad HS supportsunsynchronized breathers pinned by each HS, if the distancebetween them is large enough; hence, the breathers virtuallydo not interact. A completely different effect is displayed inFig. 5 for two broad HS, which are set closer to each other:the interaction between the breathers pinned to each HStransforms them into a stationary stable symmetric mode.It is relevant to stress that the transformation of the breatherpinned by an isolated HS into a stationary pinned state doesnot occur for the same parameters. This outcome is a genericresult of the interaction between the breathers, provided thatthe distance between them is not too large.

D. Related ModelsIn addition to the HS embedded into the medium with the uni-form nonlinearity, more specific models, in which the nonlin-earity also is concentrated at the HS, were introduced in [55]and [59]. In particular, the respective system with the doubleHS is described by the following equation:

∂u∂z

� i2∂2u∂x2

− γu

� ��Γ1 � iΓ2� − �E − iB�juj2��δ�x� L� � δ�x − L��u;(52)

where B and E are coefficients of the localized Kerr nonlin-earity and cubic loss, respectively. These settings may berealized if the nonlinear properties of the waveguides aredominated by narrow doped segments.

Although Eq. (52) seems somewhat artificial, its advantageis the possibility to find symmetric and antisymmetric pinnedmodes in an exact analytical form. Those include both funda-mental modes, with exactly two local peaks, tacked to the HS,and higher-order states, which feature additional peaks be-tween the HS. An essential finding pertains to the stabilityof such states, for which the sign of the cubic nonlinearityplays a crucial role: in the SF case [B > 0 in Eq. (52)], onlythe fundamental symmetric and antisymmetric modes, withtwo local peaks tacked to the HS, may be stable. In this case,all the higher-order multipeak modes, being unstable, evolveinto the fundamental ones. In the case of the SDF cubicnonlinearity [B < 0 in Eq. (52)], the HS pair gives rise to

multistability, with up to eight coexisting stable multipeakpatterns, symmetric and antisymmetric. The system withoutthe Kerr term (B � 0), the nonlinearity in Eq. (52) being rep-resented solely by the local cubic loss (∼E), is similar to onewith the self-focusing or defocusing nonlinearity, if the linearpotential of the HS is, respectively, attractive or repulsive, i.e.,Γ2 > 0 or Γ2 < 0 (note that a set of two local repulsive poten-tials may stably trap solitons in an effective cavity betweenthem [48]). An additional noteworthy feature of the formersetting is the coexistence of the stable fundamental modeswith robust breathers.

3. SOLITONS PINNED TO THE PT -SYMMETRIC DIPOLEA. Analytical ResultsThe nonlinear Schrödinger equation (NLSE) in the form ofEq. (17) is a unique example of a PT -symmetric system inwhich a full family of solitons can be found in an exactanalytical form [80]. Indeed, looking for stationary solutionswith real propagation constant k as u�x; z� � eikzU�x�, wherethe PT symmetry is provided by condition U��x� � U�−x�,one can readily find, for the SF and SDF signs of the nonlin-earity, respectively:

U�x� ��2k

p cos θ� i sgn�x� sin θ

cosh� ��

2kp

�jxj � ξ�� ; for σ � �1; (53)

U�x� ��2k

p cos θ� i sgn�x� sin θ

sinh� ��

2kp

�jxj � ξ�� ; for σ � −1; (54)

with real constants θ and ξ, cf. Eq. (24). The form of this sol-ution implies that Im�U�x � 0�� 0, while jumps (Δ) of theimaginary part and first derivative of the real part at x � 0are determined by the b.c. produced by the integration ofthe δ- and δ0- functions in an infinitesimal vicinity of x � 0,cf. Eq. (23):

ΔfIm�U�gjx�0 � 2γ Re�U�jx�0; (55)

Δ�

ddx

Re�U��

x�0� −2ε0 Re�U�jx�0: (56)

The substitution of Eqs. (53) and (54) into these b.c. yields

θ � arctan�γ�; (57)

which does not depend on k and is the same for σ � 1, and

ξ � 1

2��2k

p ln�σ

��2k

p� ε0��

2kp

− ε0

�; (58)

which does not depend on the PT coefficient, γ.The total power given by Eq. (29) for the localized mode is

Pσ � 2σ� ��

2kp

− ϵ0�: (59)

As seen from Eq. (58), the solutions exist at

Fig. 5. Spontaneous transformation of a pair of breathers pinned totwo hot spots, described by Eq. (51), into a stationary symmetricmode. Parameters are β � 0, ϵ � 1, γ � 1, Γ1 � 4, Γ2 � 1, and L � 4.


��2k

p> ε0 for σ � �1;��

2kp

< ε0 for σ � −1:�60�

As concerns the stability of the solutions, it is relevant to men-tion that Eq. (59) with σ � �1 and −1 satisfy, respectively,the Vakhitov–Kolokolov (VK) [97,98] and “anti-VK” [99]criteria, i.e.,

dP�1∕dk > 0; dP−1∕dk < 0; (61)

which are necessary conditions for the stability of localizedmodes supported, severally, by the SF and SDF nonlinearities;hence, both solutions have a chance to be stable. Actually, theVK and anti-VK criteria have not yet been validated for PT -symmetric systems, but it is plausible that, sharing other basicproperties with soliton families in conservative systems, local-ized modes with the unbroken PT symmetry obey thesecriteria too.

B. Numerical FindingsAs above [see Eq. (50)], the numerical analysis of the modelneeds to replace the exact δ function by its finite-width regu-larization, ~δ�x�. In the present context, the use of the Gaussianapproximation is not convenient, since, replacing the exactsolutions in the form of Eqs. (53) and (54) by their regularizedcounterparts, it is necessary, inter alia, to replace sgn�x� ≡−1� 2

Rx−∞ δ�x0�dx0 by −1� 2

Rx−∞

~δ�x0�dx0, which would be anonelementary function of x. Therefore, the regularizationwas used in the form of the Lorentzian:

δ�x� → aπ

1x2 � a2

; δ0�x� → −

2aπ

x�x2 � a2�2 ;

sgn�x� → 2π

arctan�xa

�; (62)

with 0 < a ≪ k−1∕2 [80].The first result for the SF nonlinearity, with σ � �1 (in this

case, ε0 � �1 is also fixed by scaling) is that, for given a inEq. (62), there is a critical value, γcr, of the PT coefficient,such that at γ < γcr the numerical solution features a shapevery close to that of the analytical solution in Eq. (53), whileat γ > γcr the single-peak shape of the solution transformsinto a double-peak one, as shown in Fig. 6(a). In particu-lar, γcr�a � 0.02� ≈ 0.24.

The difference between the single- and double-peak modesis that the former ones are completely stable, as was verifiedby simulations of Eq. (17) with regularization in Eq. (62), whileall the double-peak solutions are unstable. This correlation be-tween the shape and (in)stability of the pinned modes is notsurprising: the single- and double-peak structures imply thatthe pinned mode is feeling, respectively, an effective attrac-tion to or repulsion from the local defect. Accordingly, inthe latter case, the pinned soliton is unstable against sponta-neous escape, transforming itself into an ordinary freelymoving NLSE soliton, as shown in Fig. 7.

Figure 8 summarizes the findings in the plane of �a; γ� for afixed propagation constant, k � 3.0. The region of the unsta-ble double-peak solitons is a relatively narrow boundary layerbetween broad areas in which the stable single-peak solitonsexist, as predicted by the analytical solution, or no solitonsexist at all, at large values of γ. Note also that the stability areastrongly expands to larger values of γ as the regularized profile

in Eq. (62) becomes smoother, with the increase of a. On theother hand, the stability region does not vanish even for verysmall a. For the same model but with the SDF nonlinearity,σ � −1, the results are simpler: all the numerically foundpinned solitons are close to the analytical solution inEq. (54), featuring a single-peak form, and are completelystable.

A generalization of Eq. (17), including a nonlinear part ofthe trapping potential, was studied in [80] too:

i∂u∂z

� −

12∂2u∂x2

− σjuj2u − �ε0 � ε2juj2�uδ�x� � iγuδ0�x�: (63)

Equation (63) also admits an analytical solution for pinnedsolitons, which is rather cumbersome. It takes a simpler formin the case when the trapping potential at x � 0 is purelynonlinear, with ε0 � 0, ε2 > 0, Eq. (58) being replaced by

−5 0 5−0.5

0

0.5

1

1.5

2

2.5

x(a)

−5 0 5−1

−0.5

0

0.5

1

1.5

2

2.5

x(b)

Fig. 6. Comparison between the analytical solutions (solid and dot-ted blue curves show their real and imaginary parts, respectively),given by Eqs. (53), (57), and (58) with σ � �1 and ϵ0 � 1, and theirnumerically found counterparts, obtained by means of regularizationin Eq. (62) with a � 0.02 (magenta curves). The PT parameter is γ �0.20 in (a) and 0.32 in (b). In both panels, the solutions are producedfor propagation constant k � 3.

−4 −2 0 2 4 0

5

0

5

10

z

x

|u|2

Fig. 7. Unstable evolution (spontaneous escape) of the double-peaksoliton whose stationary form is shown in Fig. 6(b).


ξ � 1

2��2k

p ln

242ε2 ��

2kp

1� γ2�

��1� 8ε22k

�1� γ2�2

s 35 (64)

[it is the same for both signs of the bulk nonlinearity, σ � �1and −1, while the solution at x ≠ 0 keeps the form of Eqs. (53)or (54), respectively], with total power

Pσ�k� � 2

241� γ2

2ε2� σ

��2k

p− σ

��2k� �1� γ2�2

4ε22

s 35: (65)

Note that Eqs. (64) and (65) depend on the PT strength γ, un-like their counterparts in Eqs. (58) and (59). Furthermore,Eq. (64) exists for all values of k > 0, unlike the one givenby Eq. (58), whose existence region is limited by Eq. (60).The numerical analysis demonstrates that these solutionshave a narrow stability area (at small γ) for the SF bulkmedium, σ � �1, and are completely unstable for σ � −1.

4. GAP SOLITONS SUPPORTED BY HOTSPOT IN BRAGG GRATINGAs mentioned above, the first example of SDS supported bythe HS in lossy media was predicted in the framework of theCME (12) and (13) for the BG in a nonlinear waveguide [48]. Itis relevant to outline this original result in the present review.Unlike the basic models considered above, the CME systemdoes not admit exact solutions for pinned solitons becauseEqs. (12) and (13) do not have analytical solutions in the bulkin the presence of the loss terms, γ > 0; therefore, analyticalconsideration (verified by numerical solutions) is only pos-sible in the framework of the perturbation theory, whichtreats γ and Γ1 as small parameters, while the strength ofthe local potential, Γ2, does not need to be small. This exampleof the application of the perturbation theory is important, as itprovided a paradigm for the analysis of other models, whereexact solutions are not available either [50,54].

A. Zero-Order ApproximationA stationary solution to Eqs. (12) and (13) with γ � Γ1 � 0 andΓ2 > 0 is sought for the form that is common for quiescent BGsolitons [100,101]:

u�x; z� � U�x� exp�−iz cos θ�;v�x; z� � −V��x� exp�−iz cos θ�; (66)

where � stands for the complex conjugate, θ is a parameter ofthe soliton family, and function U�x� satisfies the followingequation:

�iddx

U � cos θ� Γ2δ�x� U � 3jUj2U − U� � 0: (67)

The integration of Eq. (67) around x � 0 yields the respectiveb.c., U�x � �0� � U�x � −0� exp�iΓ2�, cf. Eq. (23). As shownin [102], an exact soliton-like solution to Eq. (67), supple-mented by the b.c., can be found, following the pattern ofthe exact solution for the ordinary gap solitons [100–103]:

U�x� � 1��3

p sin θ

cosh��jxj � ξ� sin θ − i

2 θ� ; (68)

where offset ξ, cf. Eqs. (24), (53), and (54), is determined bythe relation:

tanh�ξ sin θ� � tan�Γ2∕2�tan�θ∕2� : (69)

Accordingly, the soliton’s squared amplitude (peak power) is

jU�x � 0�j2 � �2∕3��cos Γ2 − cos θ�: (70)

From Eq. (69), it follows that the solution exists not in thewhole interval 0 < θ < π, where the ordinary gap solitonsare found, but in a region determined by constrainttanh�ξ sin θ� < 1, i.e.,

Γ2 < θ < π: (71)

In turn, Eq. (71) implies that the solutions exist only for 0 ≤Γ2 < π (Γ2 < 0 corresponds to the repulsive HS; hence, thesoliton pinned to it will be unstable).

B. First-Order ApproximationIn the case of γ � Γ1 � 0, Eqs. (12) and (13) conserve the totalpower:

P �Z �∞

−∞�ju�x�j2 � jv�x�j�2dx: (72)

In the presence of the loss and gain, the exact evolution equa-tion for the total power is

dPdz

� −2γP � 2Γ1�ju�x�j2 � jv�x�j2�jx�0: (73)

If γ and Γ1 are treated as small perturbations, the balance con-dition for the power, dP∕dz � 0, should select a particular sol-ution, from the family of exact solutions in Eq. (68) of theconservative model, which remains, to the first approxima-tion, a stationary pinned soliton, cf. Eq. (38).

The balance condition following from Eq. (73) demandsγP � Γ1�jU�x � 0�j2 � jV�x � 0�j2�. Substituting, in the firstapproximation, the unperturbed solution in Eqs. (68) and(69) into this condition, it can be cast in the form of

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

a

γ

No solitons

Single peak solitonsDouble peak solito

ns

γ

cr

Fig. 8. Regions of the existence of stable single-peak and unstabledouble-peak PT -symmetric solitons, separated by γ � γcr�a�, in theplane of the regularization scale, a, and the gain-loss parameter, γ,for fixed k � 3.0, in the system described by Eqs. (17) and (62) withσ � �1 (the SF nonlinearity) and ε0 � 1.


θ − Γ2

cos Γ2 − cos θ� Γ1

γ: (74)

The pinned soliton selected by Eq. (74) appears, with theincrease of the relative gain strength Γ1∕γ, as a result of abifurcation. The inspection of Fig. 9, which displays �θ − Γ2�versus Γ1∕γ, as per Eq. (74), shows that the situation is quali-tatively different for Γ2 < π∕2 and Γ2 > π∕2.

In the former case, a tangent (saddle-node) bifurcation [96]occurs at a minimum value �Γ1∕γ�min of the relative gain, withtwo solutions existing at Γ1∕γ > �Γ1∕γ�min. Analysis of Eq. (74)demonstrates that, with variation of Γ2, the value �Γ1∕γ�min at-tains an absolute minimum, Γ1∕γ � 1, at Γ2 � π∕2. With theincrease of Γ1∕γ, the lower unstable solution branch, whichstarts at the bifurcation point [see Fig. 9(a)], hits the limitpoint θ � Γ2 at Γ1∕γ � 1∕ sin Γ2, where it degenerates intothe zero solution, according to Eq. (70). The upper branchgenerated, as a stable one, by the bifurcation in Fig. 9(a) con-tinues until it attains the maximum possible value, θ � π,which happens at

Γ1

γ�

�Γ1

γ

�max

≡π − Γ2

1� cos Γ2: (75)

In the course of its evolution, this branch may acquire insta-bility unrelated to the bifurcation (see below).

In the case of Γ2 > π∕2, the situation is different, as thesaddle-node bifurcation is imaginary in this case, occurringin the unphysical region θ < Γ2 [see Fig. 9(b)]. The only physi-cal branch of the solutions appears as a stable one at pointΓ1∕γ � 1∕ sin Γ2, where it crosses the zero solution, makingit unstable. However, as well as the above-mentioned branch,the present one may be subject to an instability of

another type. This branch ceases to be a physical one atpoint (75). At the boundary between the two generic casesconsidered above, i.e., at Γ2 � π∕2, the bifurcation occurs ex-actly at θ � π∕2 [see Fig. 9(c)].

The situation is different, too, in the case of Γ2 � 0 [seeFig. 1(d)], when the HS has no local potential component,and Eq. (74) takes the form of

θ

2 sin2�θ∕2� �Γ1

γ:

In this case, the solution branches do not cross axis θ � 0 inFig. 9(d). The lower branch, which asymptotically approachesthe θ � 0 axis, is unstable, while the upper one might be stablewithin the framework of the present analysis. However,numerical results demonstrate that, in the case of Γ2 � 0,the soliton is always unstable against displacementfrom x � 0.

C. Stability of Zero SolutionAs done above for the CGLEmodel with the HS [see Eqs. (47)–(49)], it is relevant to analyze the stability of the zero back-ground, and the relation between the onset of the localizedinstability, driven by the HS, and the emergence of the stablepinned mode, in the framework of the CMEs. To this end, asolution to the linearized version of the CMEs is looked for as

fu�x; z�; v�x; z�g � fA�; B�geΛz−Kx at x > 0;

fu�x; z�; v�x; z�g � fA−

; B−

geΛz�Kx; at x < 0;

with RefKg > 0. A straightforward analysis makes it possibleto eliminate the constantK and find the instability growth rate[48]:

ReΛ � −γ � �sinh Γ1�j sin Γ2j: (76)

Thus, the zero solution is subject to the HS-induced instability,provided that the local gain is strong enough:

sinhΓ1 > sinh��Γ1�cr� ≡ γ∕j sinΓ2j: (77)

Note that the instability is impossible in the absence of thelocal potential ∼Γ2. The instability-onset condition inEq. (77) simplifies in the limit when the loss and gain param-eters are small (while Γ2 is not necessarily small):

Γ1 > �Γ1�cr ≈ γ∕j sin Γ2j: (78)

As seen in Fig. 9, the pinned mode emerges at a critical pointin Eq. (78) for Γ2 ≥ π∕2, while at Γ2 < π∕2 it emergesat Γ1 < �Γ1�cr.

D. Numerical ResultsFirst, direct simulations of the conservative version ofCMEs (12) and (13), with γ � Γ1 � 0 (and Γ2 > 0) and an ap-propriate approximation for the delta function, have demon-strated a narrow stability interval in terms of parameter θ [seeEqs. (67)–(71)], close to (slightly larger than) θstab ≈ π∕2, suchthat the solitons with θ < θstab decay into radiation, while ini-tially created solitons with θ > θstab spontaneously evolve,through emission of radiation waves, into a stable one with

1 1.5 2 2.5

−0.5

0

0.5

1

1.5

2

2.5

3

θ − Γ2

Γ1 / γ

π − Γ2

1 / sin Γ2

(a)

0 1 2 3−1.5

−1

−0.5

0

0.5

θ − Γ2

Γ1 / γ

π − Γ2

1 / sin Γ2

(b)

1 1.2 1.4 1.6 1.8

−0.5

0

0.5

1

1.5

θ − Γ2

Γ1 / γ

Γ2 = 0.5π

π − Γ2

(c)

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

θ − Γ2

Γ1 / γ

Γ2 = 0

(d)

Fig. 9. Analytically predicted solution branches for the pinned gapsoliton in the BG with weak uniform loss and local gain. Shown is θ −Γ2 versus the gain strength, Γ1∕γ. (a) Γ2 � π∕4; (b) Γ2 � 3π∕4;(c) Γ2 � π∕2; (d) Γ2 � 0. In the last case, nontrivial solutions appearat point θ � 0.7442π, Γ1∕γ � 1.3801, and, at large values of Γ1∕γ, thecontinuous curve asymptotically approaches the horizontal axis. In allthe panels, the dashed lines show a formal continuation of the solu-tions in the unphysical regions, θ < Γ2 and θ > π. In panels (a)–(c), thetrivial solution, θ − Γ2 � 0, is shown by the solid line where it is stable;in the case corresponding to panel (d), all the axis θ � 0 correspondsto the stable trivial solution.


θ ≈ θstab [48,102]. This value weakly depends on Γ2. For in-stance, at Γ2 � 0.1, the stability interval is limited to0.49π < θ < 0.52π, while, at a much larger value of the localpotential strength, Γ2 � 1.1, the interval is 0.51π < θ< 0.55π. In this connection, it is relevant to mention that inthe usual BG model, based on Eqs. (12) and (13) withΓ2 � 0, the quiescent solitons are stable at θ < θ�0�cr ≈1.011 · �π∕2� [104,105].

Direct simulations of the full CME system in Eqs. (12) and(13), which include weak loss and local gain that may beconsidered as perturbations, produce stable dissipative gapsolitons with small but persistent internal oscillations, i.e.,these are, strictly speaking, breathers, rather than stationarysolitons [48]. An essential finding is that the average value of θin such robust states may be essentially larger than the above-mentioned θstab ≈ π∕2 selected by the conservative counter-part of the system (see particular examples in Table 1). Withthe increase of the gain strength Γ1, the amplitude of theresidual oscillations increases, and, eventually, the oscillatorystate becomes chaotic, permanently emitting radiation waves,which increases the effective loss rate that must be compen-sated by the local gain.

Taking values of γ and Γ1 small enough, it was checked ifthe numerically found solutions are close to those predictedby the perturbation theory in the form of Eqs. (68) and (69),with θ related to γ and Γ1 as per Eq. (74). It has been found thatthe quasi-stationary solitons (with the above-mentioned small-amplitude intrinsic oscillations) are indeed close to theanalytical prediction. The corresponding values of θ wereidentified by means of the best fit to Eq. (68). Then, for thefound values of θ and given loss coefficient γ, the equilibriumgain strength Γ1 was calculated as predicted by the analyticalformula in Eq. (74) (see a summary of the results in Table 1). Itis seen in the table that the numerically found equilibrium val-ues of Γ1 exceed the analytically predicted counterparts by∼10%–15%, which may be explained by the additional gain,which is further necessary to compensate the permanentpower loss due to the emission of radiation.

It also has been checked that the presence of the nonzeroattractive potential with Γ2 > 0 is necessary for the stability ofthe gap solitons pinned to the HS in the BG model. In additionto the analysis of stationary pinned modes, collisions betweena gap soliton, freely moving in the weakly lossy medium, withthe HS were also studied by means of direct simulations [48].

The collision splits the incident soliton into transmitted andtrapped components.

5. DISCRETE SOLITONS PINNED TO HOTSPOT IN LOSSY LATTICEThe 1D version of the discrete model based on Eq. (18), i.e.,

dun

dz� i

2�un−1 � un�1 � un−1 − 2um;n� − γun

� ��Γ1 � iΓ2� � �iB − E�junj2�δn;0un; (79)

makes it possible to gain insight into the structure of latticesolitons supported by the hot site, at which the gain and non-linearity are applied; in that case, an analytical solution isavailable [90]. It is relevant to stress that the present modeladmits stable localized states even in the case of the unsatu-rated cubic gain, E < 0 (see details below).

A. Analytical ResultsThe known staggering transformation [106,107],um�t� ≡ �−1�me−2it ~u�

m, where the asterisk stands for the com-plex conjugate, reverses the signs of Γ2 and B in Eq. (79).Using this option, the signs are fixed by setting Γ2 > 0 (thelinear discrete potential is attractive), while B � �1 or B �−1 corresponds to the SF and SDF nonlinearity, respectively.Separately considered is the case of B � 0, when the nonlin-earity is represented solely by the cubic dissipation localizedat the HS.

Dissipative solitons with real propagation constant k arelooked for by the substitution of um � Umeikz in Eq. (79). Out-side of the HS site, m � 0, the linear lattice gives rise to theexact solution with real amplitude A,

Um � A exp�−λjmj�; jmj ≥ 1; (80)

and complex λ ≡ λ1 � iλ2, localized modes corresponding toλ1 > 0. The analysis demonstrates that the amplitude at theHS coincides with A, i.e., U0 � A. Then the remaining equa-tions at n � 0 and n � 1 yield a system of four equationsfor four unknowns: A, λ1, λ2, and k; thus,

−1� cosh λ1 cos λ2 � k; −γ − sinh λ1 sin λ2 � 0;

e−λ1 sin λ2 − γ � Γ1 − EA2 � 0;

e−λ1 cos λ2 − 1� Γ2 � BA2 � k: (81)

This system was solved numerically. The stability of the dis-crete SDS was analyzed by the computation of eigenvalues formodes of small perturbations, following the known methods[107,108], and verified by means of simulations of the per-turbed evolution. (See Fig. 12 for examples of stable discretesolitons.)

In the linear version of the model, with B � E � 0 inEq. (79), amplitude A is arbitrary, dropping out from Eq. (81).In this case, Γ1 may be considered as another unknown, de-termined by the balance between the background loss andlocalized gain, which implies structural instability of the sta-tionary trapped modes in the linear model against small var-iations of Γ1. In the presence of the nonlinearity, the powerbalance is adjusted through the value of the amplitude at givenΓ1; therefore, solutions can be found in a range of values of Γ1.

Table 1. Values of the Lossa

γ θ �Γ1�num �Γ1�anal �Γ1�num−�Γ1�anal�Γ1�anal

0.000316 0.5π 0.000422 0.000386 0.09440.00316 0.595π 0.0042 0.00369 0.13730.01 0.608π 0.01333 0.01165 0.14420.1 0.826π 0.1327 0.121 0.0967

aValues of the loss parameter γ at which quasi-stationary stable pinnedsolitons were found in simulations of the CME system of Eqs. (12) and(13), by adjusting gain Γ1, for fixed Γ2 � 0.5. Values of the solitonparameter, θ, which provide for the best fit of the quasi-stationarysolitons to the analytical solution in Eq. (68), are displayed too.�Γ1�anal is the gain coefficient predicted for given γ, θ, and Γ2 by theenergy balance in the form of Eq. (74), which does not take theradiation loss into account. The right-most column shows the relativedifference between the numerically found and analytically predictedgain strength, which is explained by the necessity to compensateadditional radiation loss.


Thus, families of pinned modes can be studied, using lineargain Γ1 and cubic gain/loss E as control parameters (in theunderlying photonic lattice, their values may be adjusted byvarying the intensity of the external pump).

B. Numerical Results

1. The Self-Defocusing Regime (B = −1)The most interesting results were obtained for the unsatu-

rated nonlinear gain, i.e., E < 0 in Eq. (79). Figure 10 showsamplitude A of the stable (solid) and unstable (dashed) pinnedmodes as a function of linear gain Γ1 > 0 and cubic gain. Theexistence of stable subfamilies in this case is a noteworthyfinding. In particular, at E � 0 there exists a family of stablepinned modes in the region of 0.73 ≤ Γ1 ≤ 1.11, with the am-plitude ranging from A � 0.08 to A � 0.89. Outside of this re-gion, solutions decay to zero if the linear gain is too weak(Γ1 < 0.73) or blow up at Γ1 > 1.11. Figure 10 shows thatthe bifurcation of the zero solution A � 0 into the pinnedmode takes place at a particular value, Γ1 � 0.7286, whichis selected by the above-mentioned power-balance conditionin the linear system.

The existence of the stable pinned modes in the absence ofthe gain saturation being a remarkable feature, the stabilityregion is, naturally, much broader in the case of the cubic loss,E > 0. Figure 11 shows the respective solution branches ob-tained with the SDF nonlinearity. When the cubic loss is small,e.g., E � 0.01, there are two distinct families of stable modes,

representing broad small-amplitude (A ≤ 0.89) and narrowlarge-amplitude (A ≥ 2.11) ones. These two stable familiesare linked by an unstable branch with the amplitudes inthe interval of 0.89 < A < 2.11. The two stable branches coex-ist in the interval of values of the linear gain 0.73 ≤ Γ1 ≤ 1.13,where the system is bistable. Figure 12 shows an example ofthe coexisting stable modes in the bistability region. In simu-lations, a localized input evolves into either of these two stablesolutions, depending on the initial amplitude.

While the pinned modes may be stable against small pertur-bations under the combined action of the self-defocusing non-linearity (B � −1) and unsaturated nonlinear gain (E ≤ 0), onemay expect fragility of these states against finite-amplitudeperturbations. It was found indeed that sufficiently strongperturbations destroy those modes [90].

2. The Self-Focusing Regime (B = +1)In the case of the SF nonlinearity, B � �1, the pinned-modebranches are shown in Fig. 13, all of them being unstable with-out the cubic loss, i.e., at E ≤ 0. For the present parameters,all the solutions originate, in the linear limit, from the power-balance point Γ1 � 0.73. The localized modes remain stable,even at very large values of Γ1. Last, in the case of B � 0, whenthe nonlinearity is represented solely by the cubic gain or loss,

0 0.4 0.8 1.20

0.5

1

1.5

2

2.5

3

3.5

Γ1

A

E = 0E = −0.04

E = −0.08

E = −0.16

Fig. 10. Amplitude A of the pinned 1D discrete soliton as a functionof linear (Γ1) and cubic (E ≤ 0) gain. Other parameters in Eq. (79) areγ � 0.5, Γ2 � 0.8, and B � −1 (the SDF nonlinearity). Here, and inFigs. 11 and 13, continuous and dashed curves denote stable and un-stable solutions, respectively.

0 4 8 12 16 200

3

6

9

Γ1

A

E = 0.01E = 0.25

E = 0.66

Fig. 11. Solution branches for the discrete solitons in the case of thecubic loss (E > 0) and SDF nonlinearity (B � −1). Other parametersare the same as those in Fig. 10.

Fig. 12. Coexistence of stable small- and large-amplitude pinned dis-crete modes (top), and the corresponding evolution of nonstationarysolutions (bottom) at E � 0.01, in the bistability region. Inputs withamplitudes A � 0.3 and A � 2 evolve into the small- and large-amplitude stationary modes, respectively. The other parameters areγ � 0.5, Γ1 � 1, Γ2 � 0.8, and B � −1 (the SDF nonlinearity).

0 1 2 3 4 50

2

4

6

Γ1

A

E = −0.16

E = 0 E = 0.01

E = 0.1

E = 1

Fig. 13. Amplitude A of the discrete pinned mode as a function of thelinear gain (Γ1) and cubic loss (E) in the case of the SF nonlinearity,B � �1. The other parameters are γ � 0.5 and Γ2 � 0.8.


all the localized states are unstable under the cubic gain,E < 0, and stable under the cubic loss, E > 0.

6. CONCLUSIONThis article presents a limited review of theoretical results ob-tained in recently studied 1D and 2D models, which predict ageneric method for supporting stable spatial solitons in dissi-pative photonic media, based on the use of the linear gain ap-plied in narrow active segments (hot spots) implanted into thelossy waveguide. In some cases, the unsaturated cubic local-ized gain also may support stable spatial solitons, which is acounterintuitive finding. In view of limited length, this articlecombines a review of the broad class of such models with amore detailed consideration of selected 1D models whereexact or approximate analytical solutions for the dissipativesolitons are available. Naturally, the analytical solutions, incombination with their numerical counterparts, providedeeper insight into the underlying physical models.

A relevant issue for further development of the topic is apossibility of the existence of asymmetric modes supportedby symmetric double HS, i.e., the analysis of the spontaneoussymmetry breaking in this setting [109]. Thus far, such resultswere not demonstrated in a clear form. Another challengingextension may be the possibility to drag a pinned 1D or 2Dsoliton by an HS moving across a lossy medium.

ACKNOWLEDGMENTSI appreciate the invitation from Prof. G. Swartzlander, theEditor-in-Chief of J. Opt. Soc. Am. B, to submit this mini re-view. My work on the present topic has greatly benefited fromcollaborations with many colleagues, including P. L. Chu(deceased), N. B. Aleksić, J. Atai, O. V. Borovkova, K. W.Chow, P. Colet, M. C. Cross, E. Ding, R. Driben, W. J. Firth,D. Gomila, Y.-J. He, Y. V. Kartashov, E. Kenig, V. V. Konotop, S.K. Lai, C. K. Lam, H. Leblond, Y. Li, R. Lifshitz, V. E. Lobanov,W. C. K. Mak, A. Marini, T. Mayteevarunyoo, D. Mihalache,A. A. Nepomnyashchy, P. V. Paulau, H. Sakaguchi, V. Skarka,D. V. Skryabin, A. Y. S. Tang, L. Torner, C. H. Tsang, V. A.Vysloukh, P. K. A. Wai, H. G. Winful, F. Ye, and X. Zhang.

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Spatial solitons supported by localized gain [Invited]