Bragg management for spatial gap solitons

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Mayteevarunyoo et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. B 1957

Bragg management for spatial gap solitons

Thawatchai Mayteevarunyoo,1,* Boris A. Malomed,2 and Athikom Reoksabutr1

1Department of Telecommunication Engineering, Mahanakorn University of Technology, Bangkok 10530, Thailand2Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,

Tel-Aviv University, Tel-Aviv 69978, Israel*Corresponding author: [email protected]

Received May 20, 2010; revised July 22, 2010; accepted July 27, 2010;posted August 2, 2010 (Doc. ID 128772); published September 9, 2010

We introduce a system of nonlinear coupled-mode equations (CMEs) for Bragg gratings (BGs) where the Braggreflectivity periodically switches off and on as a function of the evolution variable. The model may be realizedin a planar waveguide with the Kerr nonlinearity, where the grating is represented by an array of paralleldashed lines (grooves), aligned with the propagation direction. In the temporal domain, a similar system can bederived for matter waves trapped in a rocking optical lattice. Using systematic simulations, we construct fami-lies of gap solitons (GSs) in this system, starting with inputs provided by exact GS solutions in the averagedversion of the CMEs. Four different regimes of the dynamical behavior are identified: fully stable, weakly un-stable, moderately unstable, and completely unstable solitons. The analysis is reported for both quiescent andmoving solitons (in fact, they correspond to untilted and tilted beams in the spatial domain). Weakly and mod-erately unstable GSs spontaneously turn into persistent breathers (the moderate instability entails a smallspontaneous change of the breather’s velocity). Stability regions for the solitons and breathers are identified inthe parameter space. Collisions between stably moving solitons and breathers always appear to be elastic.© 2010 Optical Society of America

OCIS codes: 050.2770, 230.1480, 190.5530, 060.5530.

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. INTRODUCTION AND THE MODELragg gratings (BGs) represent one basic type of mediased in photonics. They are written as periodic lattices ofefects on the surface of optical waveguides such as fibersr thin films. With spatial period � / �2 cos ��, where � ishe angle between the Poynting vector of the electromag-etic waves and the direction normal to grating (in par-icular, �=0 in the case of the fiber grating, and 0��

� /2 in planar waveguides, where the BG is realized asn array of parallel grooves), the BG provides for the reso-ant (Bragg) reflection of light and thus linear intercon-ersion between two waves copropagating at wavelength. This mechanism gives rise to the bandgap in the corre-ponding spectrum, i.e., an interval of frequencies whereinear waves cannot propagate. Fiber gratings haverawn a great deal of interest due to their important ap-lications [1] such as optical sensors and elements used inber-optic telecommunications [2] and fundamental ef-ects demonstrated by the wave dynamics in nonlinearratings [3–7].

It was predicted that the combination of the strong ef-ective dispersion near the bandgap and the Kerr nonlin-arity of the waveguide may give rise to gap solitons(GSs)a more general name for them is “BG solitons,” as theyre not necessarily centered inside the bandgap, in termsf the frequency domain) [5]. The standard model for theescription of the light transmission through the nonlin-ar BG amounts to a system of coupled-mode equationsCMEs) for amplitudes of the two waves, u�x , t� and v�x , t�,hich are linearly coupled by the Bragg reflection and in-

eract nonlinearly via the cross-phase modulation, actingn themselves through the self-phase modulation [6]. Thecaled form of the CME system is

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iut + iux + ��u�2 + 2�v�2�u + �v = 0, �1a�

ivt − ivx + ��v�2 + 2�u�2�v + �u = 0. �1b�

n the case of the fiber grating, x and t are the coordinateslong the fiber and time, while � is the Bragg reflectivity.he group velocity of light and the overall Kerr coefficientre scaled to be 1. If � is constant, it may also be replacedy 1.A well-known two-parameter family of exact solutions

or GSs is generated by Eqs. (1), with free parametershat determine the soliton’s amplitude and velocity [4,8,9]see Eqs. (4) below]. The stability of the GSs within theramework of Eqs. (1) was studied by means of the varia-ional approximation [10], and then with the help of accu-ate numerical methods [11,12]. The analysis had demon-trated that approximately half of the soliton family istable, while the remaining part is unstable. Temporalolitons in short pieces of fiber gratings ��10 cm� werereated in the experiment, launching high-power laserulses into the fiber [13–15].GSs were also predicted as spatial solitons in planar

aveguides [7,16–18] and in photonic crystals [19]. Equa-ions (2) are also relevant in that context, with t replacedy propagation coordinate z. Later, discrete GSs were pre-icted in discrete counterparts of CMEs (1) [20,21].uasi-discrete spatial GSs were created in experiments,sing arrays of semiconductor waveguides with strong cu-ic nonlinearity [22,23], arrayed photovoltaic waveguidesn LiNbO3 [24], and photonic lattices with saturable non-inearity [25–27].

Physical properties and potential applications of BGsay be vastly expanded by using gratings with various

010 Optical Society of America

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1958 J. Opt. Soc. Am. B/Vol. 27, No. 10 /October 2010 Mayteevarunyoo et al.

uperstructures (alias supergratings), which, roughlypeaking, amount to periodic variations of the Bragg re-ectivity [coefficient � in Eqs. (1)]. Superstructures haveeen investigated experimentally [28] and theoretically29]. In particular, “coupled-supermode equations” wereerived in [29] as a rather complex generalization of Eqs.1). A specific Moiré superstructure, in the form of a sinu-oidally modulated BG, was elaborated theoretically30,31] and realized in the experiment [32]. It features aarrow transmission band in the middle of the centralap, helping to create slow light in the grating. A super-tructure pattern may also be implemented in the form of

semi-discrete BG, i.e., as a waveguide with uniformonlinearity and periodically placed short segments withtrong Bragg reflectivity [33].

The form and stability of GS families in various modelsf fiber BGs with superimposed structures were studiedn detail [29,33–35]. In particular, it was found that theupergratings may open up additional bandgaps, popu-ated by solitons, and they strongly affect the stability ofhe solitons. A similar result was demonstrated in theodel of a Bose–Einstein condensate (BEC) with the self-

epulsive nonlinearity loaded into a periodic potential inhe form of an optical lattice (OL). If a periodic long-waveodulation is imposed on the OL, it gives rise to addi-

ional mini-gaps in the linear spectrum, which may alsoe populated by specific types of GSs [36].The objective of this work is to study the dynamics of

patial GSs in waveguides with the BG subjected to themanagement” in the form of a periodic modulation of theeflectivity along the propagation distance, z. If the spa-ial grating is realized, as mentioned above, as a systemf parallel grooves running along z, this simply meanshat the depth of the grooves varies periodically as a func-ion of z. In particular, it is possible to consider the modu-ation of the piecewise-constant (Kronig–Penney) type,.e., the spatial BG formed by an array of periodicallyashed grooves. The respective scaled system of theMEs is [cf. Eqs. (1)]

i�u

�z+ i

�u

�x+ ��u�2 + 2�v�2�u + ��z�v = 0, �2a�

i�v

�z− i

�v

�x+ ��v�2 + 2�u�2�v + ��z�u = 0, �2b�

here, in the case of the Kronig–Penney (“on–off”) modu-ation, the management map is defined as follows, withints period Zmap�Zon+Zoff:

��Zmap� = �� 0 � z � Zon

0 Zon � z � Zon + Zoff� . �3�

e define the map so as to scale the average reflectivity to, i.e., �=�Zon/Zmap�1, hence �=Zmap/Zon�1. Setting �1 �Zoff=0�, makes Eqs. (2) identical to the standard sys-

em (1) with the uniform grating.The difference of the present model from all previous

tudies of superstructures in BGs is that the reflectivity iseriodically modulated as the function of the evolutionalariable �z�, while in earlier works the superstructuresere represented by periodic functions of x [29,33–36].

hile the “Bragg management” of this type is obviouslyossible in the spatial domain, it is virtually impossibleor temporal solitons, which were dealt with in the earlierorks. Accordingly, the model based on Eqs. (2) and (3)elongs to the general class of management systems, inhich one or several parameters are made periodic func-

ions of the evolution variable [37]. Previously, a differentodel of the management in the spatial-domain BG sys-

em was introduced in [38], where the local nonlinearityeriodically jumped between self-focusing and self-efocusing along the propagation distance. Another physi-ally relevant example of the management, which eventu-lly leads to the same equations (2), but in the temporalomain, was experimentally realized in BEC, in the formf a “rocking” OL, i.e., the lattice which, as a whole, per-orms a periodic motion [39,40] (in the spatial domain, aounterpart of the rocking lattice was experimentally re-lized in a planar waveguide carrying a periodically un-ulating BG [41]). If the intrinsic nonlinearity of the BECs self-repulsive, which is the case in the experiment, thisetting suggests to study stability limits for GSs trappedn the rocking OL, which was done in [42]. In particular,n the limit of a weak OL, which performs the rocking mo-ion with a large amplitude, the underlying one-imensional Gross–Pitaevskii equation was reduced toqs. (2) with ��t� represented by a periodic chain of delta-

unctions.The paper is organized as follows. In Section 2, we re-

ort results of a systematic numerical analysis of “quies-ent” GSs (i.e., untilted beams in the spatial domain).our types of the dynamical behavior are identified: com-letely stable solitons; weakly unstable ones, which formobust breathers; moderately stable solitons that lose aart of their power and turn into persistent movingreathers; and completely unstable beams, quickly decay-ng into radiation. Stability borders for the solitons andreathers are identified in relevant parametric planes. Inection 3, the analysis is extended to initially “moving”olitons (actually, tilted beams), which demonstrate simi-ar types of the behavior. Their stability regions are iden-ified too. In Section 3, we also briefly consider collisionsetween stable moving solitons. Section 4 concludes theaper.

. STABILITY LIMITS FOR QUIESCENTOLITONSo look for soliton solutions in the present model, it isatural to start with initial conditions that would yieldhe usual exact GS solutions in the averaged version ofqs. (2), i.e., Eqs. (1) with �=1 [8,9]:

u0�x� = A−1�sin ��W exp�i�sech� − i�/2�, �4a�

v0�x� = − A�sin ��W exp�i�sech� + i�/2�, �4b�

here parameters � �0�� and c �−1�c�+1� deter-ine the amplitude and velocity of the GS (in fact, in the

patial domain “velocity” means a tilt of the beam carry-ng the soliton, relative to axis z):

A = ��1 − c�/�1 + c��1/4, � = �1 − c2�−1/2, �5�

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W =1 − c2

3 − c2−exp�2� + exp�− i��

exp�2� + exp�i�� 2c/3−c2

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ntegral characteristics of the soliton are its total powerwhich corresponds to the energy, in the temporal domain)nd momentum (see, e.g., [43]),

E ��−�

+�

��u�x��2 + �v�x��2�dx = 4�1 − c2�/�3 − c2�, �8�

P � i�−�

+�

�ux*u + vx

*v�dx = 4c1 − c2

7 − c2

�3 − c2�2 �sin � − � cos �� +� cos �

3 − c2 � . �9�

he power and momentum, unlike the Hamiltonian, re-ain dynamical invariants of Eqs. (2) in the presence of

he management �=��z�.The known result is that, at c=0, these GSs are stable,

s solutions to Eqs. (1), in interval 0��cr 0.505�, i.e.,lmost exactly in the half of their existence domain10–12]. At c�0, the dependence of �cr on c is very weak11].

Because the present model does not admit an analyti-al consideration, results for the existence and stability ofSs were collected from systematic simulations of Eqs.

2) with initial conditions (4). In this section, we summa-ize the findings for the quiescent GSs �c=0�, i.e., in fact,

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ig. 1. (Color online) Typical example of a stable nearly station-ry quiescent soliton for �=1.01, �=� /8, and Zmap=4.2. In thisgure and in Figs. 2–4 , panels (a) and (b) display the evolution ofhe u and v components, respectively. Panel (c) shows the evolu-ion of the soliton’s power.

or the spatial solitons which represent straight beamsunning parallel to axis z. The simulations reveal fouristinct types of the dynamical behavior: full stability,eak instability, moderate instability, and complete insta-ility, as specified below.First, stable solitons with an almost permanent shape

ere found, being similar to their counterparts given byxact solution (4) with c=0. A typical example of a com-letely stable soliton is displayed in Fig. 1. In most cases,he stable nearly stationary soliton keeps at least 99.9%f the initial power, a tiny share being lost with emittedmall-amplitude waves. Naturally, such solitons existhen the modulated model is very close to the one with

he constant reflectivity, i.e., for small values of �−1, seeq. (3).Weakly unstable solitons can be easily found too. They

earrange themselves into robust breathers, which thenemain stable indefinitely long, see an example in Fig. 2.n most cases, the breathers keep no less than 98% of thenitial power. Further, Fig. 3 demonstrates that a moder-tely strong instability originally converts the soliton intobreatherlike mode, but, after a long evolution, it spon-

aneously breaks the reflectional symmetry and startsoving slowly, and somewhat erratically, in either direc-

ion. The moving breathers, although no longer being qui-scent modes, keep their integrity until hitting an edge ofhe integration domain. The spontaneous onset of the mo-ion of a moderately unstable soliton is not affected by thenteraction with radiation waves, which were originallymitted by it and would eventually hit the soliton afterouncing back from edges of the computation domain, ashe radiation was eliminated by absorbers installed at thedges. The size of the domain was made large enough tovoid effects of absorbers on the solitons.It may seem that the spontaneous onset of motion of

he moderately unstable breathers violates the conserva-

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E

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ig. 2. (Color online) Typical example of a stable breather gen-rated by the evolution of a weakly unstable quiescent soliton,or �=1.5, �=� /10, and Z =1.
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ion of the momentum; see Eq. (9). However, Fig. 3 dem-nstrates that the actual value of the spontaneously ac-uired velocity is very small, cspont�0.008. Detailednspection of the numerical data (not shown here) demon-trates that the small momentum carried by the movingoliton is compensated by the recoil effect, i.e., by the mo-entum carried away by radiation waves emitted by theoderately unstable soliton in the course of its transfor-ation.Finally, Fig. 4 represents a typical example of the com-

lete instability of solitons, which, naturally, occurs atufficiently large values of the management period Zmapwhen the BG is switched off for a long “time”), and/orarge values of ��−1�. We stress that the solitons of all theypes would survive or decay as tightly bound states,ithout splitting into the uand v components.Stability regions for the quiescent solitons and breath-

rs, as well as the region of the existence of the moder-tely unstable solitons, are shown in Fig. 5 in the plane of� ,Zmap� for fixed �=1.01, 1.1, 1.3, and 1.7 [see Eqs. (3) forhe definitions], and in Fig. 6 in the plane of ��−1,Zmap�,or fixed �=� /10. In Figs. 5(c) and 5(d), the moderately

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x

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0.02

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0.04

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ig. 3. (Color online) Evolution of a moderately unstable ini-ially quiescent soliton, which spontaneously transforms itselfnto a moving breather, at �=1.15, �=� /8, and Zmap=2.5. Panelc) displays a detailed picture of the soliton’s motion by means ofontour plots of �u�x ,z��2 (for �v�x ,z��2 the picture is virtuallydentical).

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ig. 4. (Color online) Example of the decay of a completely un-table quiescent soliton, for �=� /8, �=1.1, and Z =3.0.
map
nstable breathers are essentially the same objects as theoderately unstable solitons defined above, i.e., sponta-eously moving pulsating modes that maintain their in-egrity (after shedding off a part of the total power in theourse of the initial evolution), and decaying breathersre the same as decaying solitons, being completely un-table objects. In Fig. 6, the vertical border between thetable solitons and breathers is a somewhat fuzzy one, astable solitons too feature very small intrinsic oscilla-ions.

. MOVING GAP SOLITONS. Stability of Moving Solitons

n experiments performed with temporal GSs in fiberratings, only moving solitons have been created thus far13–15]. As said above, in the spatial domain “moving”Ss, such as those given by Eq. (4) with c�0, actually

epresent light beams tilted with respect to the z axis.he creation of both quiescent and moving (untilted and

ilted) spatial quasi-discrete GSs was reported in arraysf parallel waveguides [23].

We tried to generate moving solitons in the presentodel by running the simulations with the input in the

orm of expressions (4) with finite values of c. A technicalroblem is that, with the available size of the simulationomain, moving solitons may reach the domain’s edgesnd hit the absorbers. This problem could be easily solvedn the following way: running the simulations in the di-ect way until the soliton would hit the absorbers, the ac-ual average velocity of the soliton c was found from theumerical data [there is a difference between c and pa-ameter c in initial conditions (4), see below]. Then, Eqs.2) were rewritten using the traveling coordinate,

y = x − cz, �10�

nstead of original x. The transformed equations take theollowing form:

i�u

�z+ i�1 − c�

�u

�y+ ��u�2 + 2�v�2�u + ��z�v = 0, �11a�

i�v

�z− i�1 + c�

�v

�x+ ��v�2 + 2�u�2�v + ��z�u = 0, �11b�

nd initial conditions (4) were transformed accordinglyoo. Then, solitons found as the numerical solutions toqs. (11) remained close to the initial position as long as

he simulations were run, allowing us to make definiteonclusions about their stability.

The results again reveal four types of the dynamical be-avior. Examples of stable moving solitons and breathersre shown in Figs. 7 and 8, respectively. As well as in thease of the quiescent solitons, the border between them isuzzy, as any soliton in the present model performs somentrinsic oscillations. A counterpart of what was defineds moderately unstable breathers in the case of the initialondition with c=0 was found here too; see an example inig. 9. In the latter case, we observe the formation ofreathers that lose a considerable part of their total

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ower, and demonstrate deviations from the steady mo-ion with small acceleration and deceleration around theverage velocity.For very small values of ��−1�—for instance, �−1

0.01—the stability border for the moving solitons withmall and large velocities, viz., c=0.1 and c=0.9, arehown in Fig. 10. In this case, stable breathers were notound as a species visibly different from the stable soli-ons. On the contrary, at larger ��−1�, such as �−1=0.2,table moving solitons with small velocities could not beenerated, but stable moving breathers were found in-tead. This result complies with the above finding that, in

0 0.1 0.2 0.3 0.4 0.5 00

1

2

3

4

5

δ/π

Zmap

Walking breathers

Uns

tabl

e

Solitons

(a)

0 0.1 0.2 0.3 0.4 0.5 00

1

2

3

4

5

δ/π

Zmap

Decaying solitons

Uns

tabl

e

Solitons

Walking breathers

(b)

ig. 5. (Color online) Stability borders for the (initially) quiescen.7 (b). In panel (a), which corresponds to very small �−1=0.01,olitons. The vertical lines at � � /2 represent the stability bordee the text.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

κ − 1

Zmap

Walking breathers

Solitons Standing breathers

Decaying solitons

ig. 6. (Color online) Stability borders for the (initially) quies-ent solitons in the plane of ��−1,Z � for fixed �=� /10.
map
he case of the zero velocity, stable breathers appear withhe increase of ��−1�; see Fig. 6. The stability borders forhis case are shown in Fig. 11(a), for c=0.2.

On the other hand, for intermediate velocities, c0.4–0.6, both stable moving solitons and breathers can

e identified. The stability borders for c=0.6 (and

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

δ/π

ap

Decaying breathers

Uns

tabl

e

Standing

breathers

Walking breathers

)

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

δ/π

ap

Decaying breathers

Uns

tabl

e

Standing

breathers

Walking breathers

)

ns in the plane of �� ,Zmap� for fixed �=1.01 (a), 1.1 (b), 1.3 (c) andagram does not have a region of completely unstable (decaying)e standard model with the uniform reflectivity, ��x��1 [10–12];

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ig. 7. (Color online) Example of a stable moving soliton gener-ted by the initial condition (4) with �=� /30, �=1.2, and velocityarameter c=0.6 in initial conditions (4) (the actual average ve-ocity produced by the simulations is c 0.667). The managementeriod is Zmap=0.6. Here and in Figs. 8 and 9, coordinate y is de-ned as per Eq. (10), with appropriate values of c.

.6

.6

Zm

(c

Zm

(d

t solitothe di

er in th

�nf

tfm

cE b=tOww

BTs

F�d

FbviF

Fv

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Fft


−1=0.2) are displayed in Fig. 11(b). At velocities c�0.6,o stable moving objects, solitons or breathers, could beound for �−1=0.2.

As said above, the average velocity c, which could be ex-racted from the numerical data, was (slightly) differentrom the value of c in the initial conditions (4). For stableoving solitons, Fig. 12 shows the residual velocity,

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ig. 8. (Color online) Example of a stable moving breather with=� /30, �=1.2, c=0.6, and Zmap=1.2. The average velocity pro-uced by the simulations is c 0.658.

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ig. 9. (Color online) Example of moderately unstable movingreather with �=� /10, c=0.6, Zmap=1.5, and �=1.2. The averageelocity produced by the simulations is c 0.538 (note that c�cn this case, on the contrary to c�c in the cases displayed inigs. 7 and 9).

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

2.5

3

δ/π

Zmap

Moving solitons

Unstable solitons

c = 0.1c = 0.9

ig. 10. (Color online) Stability borders for moving solitons withelocities c=0.1 and c=0.9, at ��−1�=0.01.

residual� c−c, versus ��−1� and Zmap for fixed �=� /10.xcept for the point with the negative value, cresidual−0.062, which corresponds to the moderately unstable

reather displayed in Fig. 9 (for Zmap=1.5, �=1.2 and �� /10, c=0.6), all other data points, with cresidual�0, per-

ain to stable solitons and weakly unstable breathers.ne can see that the residual velocity of the stable andeakly unstable modes strongly depends on ��−1�, andeakly depends on the management period, Zmap.

. Collisions between Stable Moving Solitonshe availability of stable moving solitons suggests a pos-ibility to consider collisions between them. In the stan-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

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(a)

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Unsteadily moving breathers

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(b)

ig. 11. (Color online) Stability borders for the moving solitonsith �−1=0.2: (a) c=0.2, (b) c=0.6.

0 0.5 1 1.5 2−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

(κ−1), Zmap

c resi

dual

(κ−1)Z

map

c = 0.6

c = 0.1

ig. 12. (Color online) Residual velocity (defined in the text) asunctions of �−1 and Zmap, at fixed Zmap=1 and �=1.2, respec-ively. In both cases, �=� /10.

dGlefm

sf[st

mwtpapstfi

4Ittphtrwpsmdttd(cmTsbSim

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R

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Fi=


ard model based on CMEs (1), collisions between movingSs were studied in detail, by means of systematic simu-

ations [44,45]. The collisions were also studied in a gen-ralized model that contains additional terms accountingor the dispersion of the Bragg reflectivity [46]. In theodel of the superstructure represented by a chain of

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ig. 13. (Color online) Collision between stable solitons movingith velocities c= ±0.6, the other parameters being �−1=0.2,map=0.5, and �=� /10.

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ig. 14. (Color online) Collision between two stable moving soli-ons at �−1=0.2 and Zmap=0.5. The initial parameters of the leftnd right solitons are, respectively, c=0.6, �=� /10, and c=−0.5,=� /30.

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ig. 15. (Color online) Collision between stable breathers mov-ng with velocities c= ±0.6, the other parameters being �−10.2, Z =1.2, and �=� /30.
map
hort BG segments, against the background of the uni-orm Kerr nonlinearity, collisions were studied in detail in33]. In all these models, regions of quasi-elastic andtrongly inelastic collisions, that might lead to merger ofhe colliding solitons, were identified.

In all cases considered in the framework of the presentodel, collisions between stable solitons and breathersere elastic. As shown in Figs. 13 and 14, this is true for

he collisions between identical solitons moving with op-osite velocities, ±c, and for soliton pairs with differentmplitudes and/or velocities. Figure 15 additionally dis-lays a typical example of the elastic collision betweentable breathers. The elasticity of the collisions attests tohe fact that the solitons and breathers that were identi-ed as stable ones are indeed very robust objects.

. CONCLUSIONn this work we have studied families of GSs (gap soli-ons) in the framework of the CME (coupled-mode equa-ion) system in which the Bragg reflectivity was made aiecewise-constant function of the evolution variable. Weave adopted the management map (3), with the reflec-ivity periodically switching off and on. The model may beealized in a straightforward way in a nonlinear planaraveguide, with the grating represented by an array ofarallel dashed lines (grooves). In the temporal domain, aimilar CME system was derived as a limit form of theodel for BEC loaded into a rocking optical lattice. By

int of systematic simulations, which made use of the ini-ial conditions corresponding to the exact GS solutions inhe averaged version of the model, we have identified fourifferent types of the dynamical behavior of the solitonsfully stable, weakly unstable. moderately unstable, andompletely unstable). This was done for the quiescent andoving solitons (actually, untilted and tilted ones) alike.he weakly and moderately unstable solitons turn them-elves into persistent breathers (in the latter case, thereather features an erratic motion with a small velocity).tability regions for the solitons and breathers have been

dentified. It was concluded that collisions between stableoving solitons and breathers are always elastic.

CKNOWLEDGMENTShe work of T. M. was supported, in a part, by the Tel Avivniversity through a postdoctoral fellowship granted by

he Pikovsky–Valazzi Foundation and by the Thailandesearch Fund under grant RMU5380005. B.A.M. appre-iates hospitality of the Department of Telecommunica-ion Engineering at the Mahanakorn University of Tech-ology (Bangkok, Thailand).

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Bragg management for spatial gap solitons