solitons in a linearly coupled system with separated dispersion and nonlinearity

CHAOS 15, 037108 �2005�

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Solitons in a linearly coupled system with separated dispersionand nonlinearity

Arik Zafrany, Boris A. Malomed,a� and Ilya M. MerhasinDepartment of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering,Tel Aviv University, Tel Aviv 69978, Israel

�Received 25 January 2005; accepted 19 February 2005; published online 21 October 2005�

We introduce a model of dual-core waveguide with the cubic nonlinearity and group-velocitydispersion �GVD� confined to different cores, with the linear coupling between them. The modelcan be realized in terms of photonic-crystal fibers. It opens a way to understand how solitons aresustained by the interplay between the nonlinearity and GVD which are not “mixed” in a singlenonlinear Schrödinger �NLS� equation, but are instead separated and mix indirectly, through thelinear coupling between the two cores. The spectrum of the system contains two gaps, semi-infiniteand finite ones. In the case of anomalous GVD in the dispersive core, the solitons fill the semi-infinite gap, leaving the finite one empty. This soliton family is entirely stable, and is qualitativelysimilar to the ordinary NLS solitons, although shapes of the soliton’s components in the nonlinearand dispersive cores are very different, the latter one being much weaker and broader. In the caseof the normal GVD, the situation is completely different: the semi-infinite gap is empty, but thefinite one is filled with a family of stable gap solitons featuring a two-tier shape, with a sharp peakon top of a broad “pedestal.” This case has no counterpart in the usual NLS model. An extendedsystem, including weak GVD in the nonlinear core, is analyzed too. In either case, when thesolitons reside in the semi-infinite or finite gap, they persist if the extra GVD is anomalous, andcompletely disappear if it is normal. © 2005 American Institute of Physics.�DOI: 10.1063/1.1894705�

The first thing that everyone knows about solitons is thatthey are supported by balance between linear dispersionand nonlinearity, a paradigm for which is provided bythe nonlinear Schrödinger (NLS) equation. With an ob-jective to understand the interplay between the second-order dispersion and cubic nonlinearity from anotherperspective, we introduce a simple but new model, whichis constructed as a system of two equations coupled bylinear terms, with the nonlinearity confined to one equa-tion, and the dispersion to the other. The linear spectrumof this model features a semi-infinite gap (a spectral re-gion where linear wave solutions do not exist, and there-fore solitons are possible), essentially the same as in theusual NLS equation, and, in addition, a finite gap. It isknown from studies of other models that a finite gap mayhost solitary-wave solutions of a different type, the so-called gap solitons. We find that, in the present model, onegap is always filled with solitons, and the other one isempty. Namely, for negative dispersion (known as anoma-lous dispersion in optics), which is the case when the or-dinary NLS solitons exist, the semi-infinite gap is filled,while the finite one is empty. The most interesting casecorresponds to positive (normal) dispersion, when theusual NLS equation does not give rise to (bright) solitons:in this case, the finite gap in our model is filled with gap-soliton solutions. The shape of the two-component soli-tons found in the model is quite different from that of theclassical solitons, with a very narrow and tall component

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in the nonlinear mode, and a broad and low componentin the dispersive mode. The solitons in the finite gap, ex-isting in the case of the normal dispersion, feature anespecially interesting two-tier shape, with a narrow peaktowering above a broad small-amplitude socle (“pedes-tal”). We show that both soliton families, in the semi-infinite and finite gaps, are completely stable againstsmall perturbations. We also investigate an extendedmodel, with weak dispersion added to the nonlinear sub-system; in that case, the soliton solutions persist if theextra dispersion is negative. The model proposed in thiswork can be realized in photonic-crystal fibers with theintrinsic Kerr nonlinearity.

I. INTRODUCTION

It is commonly known that solitons are supported, inintegrable and nonintegrable systems alike, by balance be-tween the linear group-velocity dispersion �GVD� and non-linear self-compression of the wave field�s�.1 A paradigmaticmodel of the nonlinear wave propagation which demon-strates this mechanism in action is the nonlinear Schrödinger�NLS� equation, in the case when the GVD is anomalous, interms of the propagation of optical signals in material media.A standard form of the corresponding NLS equation is

iuz + �1/2�u�� + �u�2u = 0, �1�

where the subscripts stand for the partial derivatives, u�z ,��is the local amplitude of the electromagnetic wave, z and t
are the propagation distance and time, �� t−z /V0 �V0 is the
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037108-2 Zafrany et al. Chaos 15, 037108 �2005�

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group velocity of the carrier wave�, and the cubic term ac-counts for the Kerr effect �a nonlinear correction to the ef-fective refractive index of the medium�. Equation �1� yields afamily of stable soliton solutions,

usol = � sech�� − cz��exp�ic� + �i/2��2 − c2�z� , �2�

with an arbitrary amplitude � and inverse-velocity shift c. Inthe reference frame with c=0, the propagation constant, k��2 /2�0, of the soliton solution belongs to the semi-infinite gap, while the remaining semi-infinite band, k�0, isfilled by linear waves �hence any localized pulse with k�0decays into the radiation�.

A different situation occurs in optical fibers with thesame Kerr nonlinearity if the dispersion is induced by theBragg grating �BG�, i.e., a periodic array of defects written inthe fiber’s cladding. The BG of period a gives rise to theresonant linear coupling between left- and right-travelingwaves, u�z , t� and v�z , t�, with the wavelength �=2a /n,where n is an integer �usually, n=1�. The BG written on thenonlinear fiber is described by the well-known equations �inwhich the group velocity of light and BG-induced linear cou-pling coefficient are normalized to be 1�,2

iut + iuz + ��u�2/2 + �v�2�u + v = 0,

ivt − ivz + ��v�2/2 + �u�2�v + u = 0. �3�

This model neglects intrinsic dispersion of the fiber, as theartificial BG-induced dispersion is much stronger �takinginto regard the intrinsic dispersion drastically alters solitonsolutions of the model,3 although this would only be ob-served in unrealistically long fiber gratings�.

Linearizing Eq. �3� and looking for a solution as �u ,v��exp�iqz− i�t�, one concludes that, unlike the NLS equa-tion, the BG model supports only a finite gap in the spec-trum, �2�1. Although Eq. �3� is not integrable, unlike theNLS equation, a family of exact gap-soliton solutions isavailable in this model.2,4 The solutions with the zero veloc-ity precisely fill out the gap. While Eq. �3� does not featureGalilean or Lorentzian invariance, exact solutions for soli-tons moving with any velocity c, up to �c�=1, were foundtoo.2,4

The stability of these BG solitons is a nontrivial prob-lem. First of all, they all satisfy the known Vakhitov–Kolokolov �VK� criterion, dE /d��0, which is a necessarybut �in the general case� not sufficient stability condition.5

Here, the soliton’s energy is E=�−�+��u�x��2+ �v�x��2�dx, and

� is the frequency of the solution. For the stability of theNLS solitons, this criterion is actually sufficient. However,the BG solitons may be destabilized by perturbations associ-ated with complex stability eigenvalues, which, unlike theircounterparts pertaining to real eigenvalues, are not taken intoregard by the VK criterion. As a result, it was found, first bymeans of the variational approximation6 and later in a rigor-ous form, based on numerical computation of the full set ofeigenvalues,7 that, within their existence region, 0��cos−1 ��, only slightly more than half of the solutions
are stable—viz., ones in the interval

0 � � � �cr 1.01 �/2� . �4�

Additionally, it was found that the stability border for themoving solitons, �cr�c�, varies very weakly with c, up to �c�=1.7

More sophisticated nonlinear-optical models may giverise to two gaps, an example being a BG integrated with aperiodic system of thin �subwavelength� layers of a materialresonantly interacting with light �two-level atoms tuned tothe resonance�. Depending on values of parameters, one orboth gaps are partly filled by stable solitons in the lattersystem.8

In this work, our objective is to introduce a seeminglysimple but nevertheless new �to the best of our knowledge�model of a dual-core waveguide with completely separatedGVD and nonlinearity, so that one core, which carries amode u, is nonlinear, and the other one, carrying a mode v, isdispersive, cf. Eq. �1�,

iuz + �u�2u + v = 0, �5�

ivz + qv + �D/2�v�� + u = 0. �6�

The model assumes that the cores are linearly coupled, withthe coupling coefficients normalized to be 1. In Eq. �6�, realparameters q and D are the phase-velocity mismatch betweenthe cores, and the GVD coefficient that we may always set tobe +1 or −1, which corresponds to the anomalous or normaldispersion, respectively. We note that group-velocity terms,such as ic1u� in Eq. �5� and ic2v� in Eq. �6� �with some realcoefficients c1 and c2�, can be removed: the former term bythe shift of the references frame, �→�−c1z, and the latterone by the phase transformation, v→v exp�ic2� /D�. There-fore, these terms are not included.

Equation �5� assumes that the dispersion in the nonlinearcore is completely negligible. In real systems, some residualdispersion is always present. In that case, Eq. �5� is to bereplaced by the following equation, with a small dispersioncoefficient �:

iuz + ��/2�v�� + �u�2u + v = 0. �7�

The model with Eq. �5� substituted by Eq. �7� will also beconsidered in the following.

The objective of the study of the system based on Eqs.�5� and �6� is to understand what kind of solitons can besupported by the interplay of the separated nonlinearity andGVD of either sign. Previously, a somewhat similar model ofa dual-core BG was introduced in Ref. 9, with the BG writ-ten on the linear core, and the nonlinearity concentrated inthe other one. In that model, the main result was finding avery dense family of embedded solitons, i.e., ones existingnot in the band gap, but rather embedded in the band ofradiation waves �solitons of this type are not possible insimple models, such as Eqs. �1� and �3�, but may exist insufficiently complex multicomponent systems.10�. Anotherrelated model was introduced in Ref. 11. It assumed a dual-core waveguide with the same nonlinearity in both cores, andopposite signs of the GVD in them; in particular, one GVD
coefficient could be zero.

037108-3 Separated dispersion and nonlinearity Chaos 15, 037108 �2005�

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In principle, the model with the completely or nearlyseparated nonlinearity and GVD can be realized in a dual-core optical fiber with strongly asymmetric cores.11 How-ever, a more promising medium is a photonic-crystal fiberwith two embedded cores, one �nonlinear� made of solidsilica, and one hollow. Then, the necessary GVD can be de-signed through the structure of the lattice of thin holes run-ning parallel to the fiber’s core. In fact, fabrication of two-core photonic-crystal fibers has already been reported.12

The same model, based on Eqs. �5� and �6�, admits an-other physical interpretation—in the spatial domain, if thevariable � in Eq. �6� is replaced by the transverse coordinatex. In that case, we are dealing with two parallel planarwaveguides, nonlinear and linear ones. The second deriva-tive accounts for the diffraction in the transverse direction,and the absence of the diffraction in the nonlinear waveguideimplies that it is actually not a solid one, but is built as anarray of one-dimensional “ribs.”

A peculiarity of the present model is that it gives rise toa combination of two gaps, finite and semi-infinite ones. As

FIG. 1. A typical example of the soliton found in the semi-infinite gap, fork=5 and q= +0.8, in the model with the separated dispersion and nonlinear-ity in the case of anomalous dispersion, D= +1. Panels �a� and �b� displaythe nonlinear- and dispersive-mode components of the soliton.

concerns solitons existing in these gaps, the basic result re-


ported in the following is that they are found solely in thesemi-infinite gap in the case of the anomalous GVD �D=+1 in Eq. �6��, and solely in the finite gap in the oppositecase of the normal dispersion, D=−1. In either case, theentire family of the solitons is found to be stable in directsimulations. We will also study the structural stability of thesolitons, i.e., how they are effected by adding weak GVD tothe originally dispersionless nonlinear mode. Obviously,small GVD is always present in a strongly nonlinear core ofany real-world system.

The paper is organized as follows. In Sec. II, we con-sider the linear spectrum of the model. Basic results, i.e.,soliton families and their stability in the models with theanomalous or normal GVD, are presented in Sec. III. SectionIV reports additional results, such as the above-mentionedstructural stability and attempts to create “moving” solitons.Section V concludes the paper.

II. THE GAP STRUCTURE OF THE SYSTEM

FIG. 2. The same as in Fig. 1, but for q=−0.8.

Stationary solutions to Eqs. �5� and �6� are sought for as



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�u,v� = �U��,V��eikz, �8�

where k is a real propagation constant, and real functionsU�� and V�� obey

U3 + V = kU, qV + �D/2�V� + U = kV . �9�

The stationary solutions are characterized by the value of thetotal energy, which is a dynamical invariant of Eqs. �5� and�6� �the Bragg-grating model �3� has the same dynamicalinvariant�,

E � Eu + Ev = −�

+�

�U2�� + V2��d� . �10�

Looking for a solution to the linearized version of Eq.�9� in the form of plane waves, �U ,V�= �A ,B�exp�ikx− i��

FIG. 3. �a� A typical example of the dependence of the soliton’s energy onthe propagation constant k for the soliton family in the semi-infinite gap, forq=−0.8, in the case of anomalous dispersion, D= +1. �b� The share of theenergy in the nonlinear-wave component vs k, in the same soliton family.

with real �, we arrive at the dispersion relation for them,


k = −1

2�1

2D�2 − q� ± 1

4�1

2D�2 − q�2

+ 1. �11�

Straightforward consideration of the spectrum defined by Eq.�11� demonstrates that, in the case of the anomalous GVD�D= +1�, it gives rise to finite and semi-infinite gaps,

−1

2� 4 + q2 − q� � k � 0,

1

2� 4 + q2 + q� � k � � ,

�12�

and in the case of the normal GVD �D=−1�, the semi-infiniteand finite gaps are

− � � k �1

2� 4 + q2 − q�, 0 � k �

1

2� 4 + q2 + q�

�13�

�the dispersion relation �11� is not displayed here in a graphi-cal form, as all what we actually need from it is the exact

FIG. 4. The full widths at half-maximum for the nonlinear �a� and disper-sive �b� components of the soliton family in the semi-infinite gap, vs thepropagation constant, for D= +1 and q=−0.8.

form of the band gaps, as given by the above expressions�.



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Note that both gaps �12� are broader than their counterparts�13� for q�0, and vice versa for q�0.

It is easy to understand how the spectrum is modified ifsmall GVD of the nonlinear mode is taken into regard, i.e.,Eq. �5� is replaced by Eq. �7� with small �. If both the mainand additional GVDs are anomalous, i.e., D= +1 and ��0,or both are normal, i.e., D=−1 and ��0, the finite gap willbe closed down, but the semi-infinite one survives. On theother hand, if the the sign of the extra GVD is opposite tothat of the main dispersive term, i.e., D��0, the modifiedspectrum maintains the finite gap, but closes down the semi-infinite one.

We also note that, in the case of �=0, Eq. �11� yields�2=2�Dk�−1�1+qk−k2�, which implies that, inside both thefinite and semi-infinite gaps, �2 takes real negative values.This fact suggests a possibility of the existence of ordinarysolitons with monotonically decaying tails. On the otherhand, in the model which has dispersion in both linearlycoupled components, �2 is sometimes complex in a part ofthe �single� gap supported by such a model, which meansthat the corresponding gap solitons may feature tails which

11

FIG. 5. A typical example of the evolution of the nonlinear �a� and disper-sive �b� components of a perturbed soliton belonging to the semi-infinite gapin the model with anomalous dispersion, D= +1. The unperturbed soliton isthe same as in Fig. 1. The relative size of the initial amplitude perturbationis 5%.

decay with oscillations.


One can also look for solitons moving relative to thereference frame in which Eqs. �5� and �6� are written. Thisimplies looking for a soliton solution in the form of

FIG. 6. A typical example of the nonlinear-mode �a� and dispersive-mode�b� components of a gap soliton, found in the finite gap of the system withnormal dispersion, D=−1 in Eq. �6�. �c� The distribution of the total localpower, P=U2��+V2��, in the soliton. In this case, k=0.4 and q=0.8.



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�u,v� = �U�� − cz�,V�� − cz��eikz, �14�

where c is the soliton’s inverse-velocity shift �cf. Eq. �8��.The consideration of the system obtained by the substitutionof Eq. �14� in Eqs. �5� and �6� shows that, in terms of thevariables �z ,�� t−cz�, no true gap �finite or semi-infiniteone� is possible, hence we do not expect the existence ofmoving solitons in the present model. The situation can bechanged by adding the third-order dispersion to the v mode:as was shown in a different model �a system of coupled bulkand surface modes co-guided by a hollow core in a photonic-crystal fiber�,13 in this case a true finite gap is possible formoving solitons too, and such stable solitons exist indeed inthe full nonlinear system.

III. SOLITONS AND THEIR STABILITY

A. The case of anomalous dispersion

Soliton solutions of stationary equations �9� were looked

FIG. 7. A typical example of the dependencies of the total energy �a� andshare of the energy in the nonlinear-mode component �b� on the propagationconstant for the gap-soliton family in the system with normal dispersion, forq=0.8.

for in the numerical form by means of the relaxation method.


As a result, it was found that solitons exist in the semi-infinite gap if the system’s dispersion is anomalous, D= +1,while the finite gap remains completely empty in this case�see Eq. �12��. This result does not depend on the size andsign of the mismatch parameter q. Typical examples of thesolitons found in the semi-infinite gap are shown in Figs. 1and 2. It is obvious that the shape of the soliton in the dis-persive mode �v� is much smoother than in the nonlinear one�u�, due to the presence of the second derivative. Neverthe-less, the shape of the solitons is strictly smooth in both com-ponents; in particular, there is no true cusp at the tip of thedispersive component �see Figs. 1�b� and 2�b��.

The soliton family was found to completely fill the semi-infinite gap, in the case of D= +1. A global characteristic ofthe family is the soliton’s energy, defined as per Eq. �10�, asa function of the propagation constant k. Note that this de-pendence provides for a necessary stability condition, in theform of the above-mentioned VK criterion,5 dE /dk�0. Ac-tually, the VK criterion may guarantee only the stabilityagainst small perturbations with real eigenvalues, ignoringpossible instability with complex eigenvalues. Full stabilityof the solitons was tested by means of direct numerical simu-lations, see the following.

In addition to E�k�, a characteristic of the soliton familyis the share of the total energy in the nonlinear �u� mode,Eu / �Eu+Ev�, also as a function of k. Both dependencies aredisplayed, for a typical soliton family, in Fig. 3. As is seen,the dependencies are monotonic, and a larger part of energyis carried by the nonlinear mode.

Besides the energy and its distribution between the twocomponents, another essential parameter of the soliton is itswidth, which we define as the standard full width at half-maximum, separately in the u and v modes. The dependen-cies of the widths versus k are shown in Fig. 4. It is seenfrom these plots that �as could be naturally expected� thesoliton becomes very broad, and its amplitude vanishes, asone approaches the edge of the semi-infinite gap �for thisreason, the plots in Figs. 3 and 4 do not start exactly at thegap’s edge, where it is difficult to accurately compute thecharacteristics of the soliton�. In the opposite limit, k→�,the soliton becomes very narrow, and its amplitude and en-ergy become very large.

Stability of the solitons was tested in direct simulations,by adding an arbitrary perturbation at the initial point, z=0.Systematic simulations have unequivocally demonstratedthat all the solitons are stable in the semi-infinite gap �which,in particular, complies with the prediction of the VK crite-rion, as seen from Fig. 3�a��. A typical example of the evo-lution of a perturbed soliton �with an initial perturbationwhich, actually, was not very small� is shown in Fig. 5. Anadditional illustration of the stability of the soliton againstquite strong perturbations will be provided by Fig. 12, whichis displayed in the following in a different context.

B. Solitons in the system with normal dispersion

The above-found soliton family is, as a whole, qualita-tively similar to the family of classical solitons �2� in the
NLS equation �1� with the anomalous GVD, even if the


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shape of the solitons is very different from that in the NLSequation. In the case of normal GVD, the single NLS equa-tion does not support bright solitons. A drastic difference ofthe present model, with the separated nonlinearity and dis-persion, is that solitons are found too in the case of the nor-mal dispersion, D=−1 in Eq. �6�. While the semi-infinite gapis empty in this case, the finite one is completely filled by�numerically found� solitons �see Eq. �13��. We consider thegap-soliton family, found in the case of the normal GVD, asthe most significant result reported in the present work.

An example of the gap soliton found in the normal-GVDmodel is displayed in Fig. 6. Difference from the solitons inthe model with anomalous dispersion is obvious from com-parison with Figs. 1 and 2: in terms of the local power,�u��2+ �v��2, the gap soliton features a two-tier shape,with a tall peak towering above a relatively broad socle�“pedestal”�, see Fig. 6�c�. This result is noteworthy, as ordi-nary gap solitons in the model of the fiber Bragg grating �3�do not show any similar structure. Actually, in the caseshown in Fig. 6�c� the socle is very low and barely visible; itwill be much more pronounced if weak anomalous disper-

FIG. 8. Dependencies of the full width at half-maximum of the nonlinear-mode �a� and dispersive-mode �b� components of the gap soliton in thesystem with normal dispersion �D=−1�, for q=0.8.

sion is added to the nonlinear core, see Figs. 11�c� and 12.


As well as in the case of the anomalous GVD, globalcharacteristics of the soliton family are presented by the de-pendencies of E�k� and Eu / �Eu+Ev� vs k and, additionally,by plots showing the widths of both components of the soli-ton versus k, see Figs. 7 and 8. In the figures, the dependen-cies are shown across the entire finite gap. As can be con-cluded from the figures, the amplitude and energy of thesoliton vanish as one approaches the left edge of the gap, k=0 �for this reason, the plots do not reach the point k=0, asit is difficult to identify the soliton with a very small ampli-tude very close to this point�, and the amplitude and energydiverge as one approaches the right edge of the finite gap�see Eq. �13��. It is quite noteworthy that, as seen from Fig.8, the soliton’s widths remain finite near both edges, and thegeneral dependence of the widths on k is inverse to that inthe case of the anomalous GVD, cf. Fig. 4. Note also that theshare of the energy in the dispersive-mode component of thesoliton may be much greater than in the case of the anoma-lous GVD, cf. Figs. 7�b� and 3�b�.

Figure 7�a� suggests that the entire family of the gapsolitons in the system with normal GVD should be stableaccording to the VK criterion. Direct simulations confirm

FIG. 9. A typical example of stable evolution of the nonlinear-mode �a� anddispersive-mode �b� components of a perturbed gap soliton in the case ofnormal dispersion. The corresponding unperturbed soliton is the one shownin Fig. 6.

that, indeed, all the gap solitons are stable, see an example



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in Fig. 9. This result should be compared to the fact that onlya part of the standard gap-soliton family generated by Eq. �3�is stable, see Eq. �4�. On the other hand, a caveat is thatdirect simulations can miss some very weak instabilityagainst oscillatory perturbations.

IV. STRUCTURAL STABILITY OF THE SOLITONS

The above-mentioned results presented for the dual-coremodel with strictly separated nonlinearity and GVD. As real-world systems would have some amount of nonlinearity anddispersion in either core, it is necessary to investigate thestructural stability of the solitons, in this sense. In fact, it isreally necessary to check the stability of the solitons againstthe addition of weak dispersion to the nonlinear core, whichis accounted for by the coefficient � in Eq. �7�. This analysisshould supplement the above-noted conclusions about thedynamical stability of solitons in both the anomalous- and

FIG. 10. A typical example of a soliton in the semi-infinite gap of thesystem with anomalous GVD in the dispersive mode, D= +1, in the casewhen weak anomalous dispersion was added to the nonlinear subsystem��=0.1�. As before, here, and in Fig. 11, �a� and �b� show, respectively, thesoliton’s components in the �formerly� nonlinear and dispersive modes, i.e.,u and v. The parameters are k=5 and q=0.8 �the same as in Fig. 1�.

normal-GVD systems.


As was shown earlier, the addition of weak normal dis-

FIG. 11. A typical example of a soliton in the finite gap of the system withnormal GVD in the dispersive mode, D=−1, in the case when weak anoma-lous dispersion was added to the nonlinear subsystem ��=0.1�. As in Fig. 6,�a�, �b�, and �c� show, respectively, the soliton’s components in the �for-merly� nonlinear and dispersive modes, i.e., u and v, and the distribution ofthe local power across the soliton. The other parameters are k=0.4 and q=0.8, i.e., the same as in Fig. 6.

persion �corresponding to ��0 in Eq. �7�� to the nonlinear



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core would close down, respectively, the semi-infinite gap inthe system with the anomalous GVD in the dispersive core,and the finite gap in the opposite case of the dispersive corewith the normal GVD. This simple conclusion suggests thatthe soliton family in the semi-infinite gap in the former case,and in the finite gap in the latter case is destroyed by theaddition of weak normal dispersion to the nonlinear core ineither case, thus admitting no solitons to survive in the sys-tem. This prediction was indeed confirmed by direct numeri-cal solutions, which could not turn up any soliton solution insuch a case.

On the other hand, the same argument suggests that, ineither case as well �anomalous or normal GVD in the disper-sive core�, the soliton families may survive if weak anoma-lous GVD ��0� is included in Eq. �7� for the nonlinearsubsystem. This expectation was also confirmed by numeri-cal results, which readily reveal solitons in this case. Ex-amples of the thus found solitons in the semi-infinite andfinite gaps are shown in Figs. 10 and 11, respectively. As isseen, a difference from the above-found solitons in the ide-alized model is that they �quite naturally� get smoothed bythe dispersion in the former nonlinear core. Direct simula-tions have demonstrated that all the smoothed solitons, foundwith ��0, remain dynamically stable.

A noteworthy feature which is strongly enhanced by theweak anomalous dispersion added to the nonlinear core is thetwo-tier structure of the solitons in the finite gap, as is obvi-ous from Fig. 11. The socle in the gap soliton, in the modelincluding the weak anomalous dispersion in the nonlinearsubsystem, can be still more salient at other values of theparameters, see an example in Fig. 12. However, we havenever seen an example of a gap soliton in which the distri-bution of the local power would feature side peaks, separatefrom the central one.

It was also concluded earlier in this paper that movingsolitons are impossible in the present model. This predictiontoo was checked against direct simulations. To this end, both

FIG. 12. The same as in Fig. 11�c�, but for k=1.2. In this example, thetwo-tier structure of the gap soliton is strongly pronounced.

u and v components of a stable stationary soliton were mul-


tiplied by the “shove” factor, exp�−i��, with a constant real�. As a result, it was found that there is a critical value, �cr,of the shove strength, such that the soliton does not set inmotion, but relaxes back into a state with zero velocity, if��cr. On the other hand, the push leads to complete de-struction of the soliton if ��cr.

For typical values of other parameters considered earlier,�cr is nearly constant, taking values between 0.9 and 1.1. Anexample of stable relaxation of the initially shoved solitoninto a stable quiescent one, when � is close to the criticalvalue, is displayed in Fig. 13.

V. CONCLUSION

In this work, we have introduced a model of dual-corewaveguides with complete separation of the nonlinearity andGVD. Each ingredient was confined to one of two cores,with the linear coupling between them. The model can berealized in optics: either in photonic-crystal fibers in the tem-poral domain, or in a system including a solid linear planarwaveguide and an array of nonlinear rib waveguides,parallel-coupled to it �but not to each other�, in the spatial

FIG. 13. Relaxation of the nonlinear-mode �a� and dispersive-mode �b�components of a soliton to which the “shove” was applied, in the form of thefactor exp�−i��, with �=−0.9. The unperturbed soliton is the same as inFig. 1 �i.e., it belongs to the semi-infinite gap�.

domain. The model offers a possibility to understand how



D

solitons are sustained by the interplay between the Kerr �cu-bic� nonlinearity and GVD which are not “mixed,” as in thesingle NLS equation, but are separated and interact indi-rectly, via the linear coupling between the two cores. Themodel gives rise to semi-infinite and finite gaps in its spec-trum. In the case of the anomalous GVD in the dispersivecore, the solitons fill the semi-infinite gap, leaving the finiteone empty. This soliton family is stable, and is qualitativelysimilar to the ordinary NLS solitons �2�, although the shapesof the nonlinear-mode and dispersive-mode components ofthe soliton are very different, the former one being muchtaller and narrower. In the case when the GVD is normal, thesituation was found to be drastically different: while thesemi-infinite gap is empty, the finite one is filled with a com-pletely stable family of solitons that have a specific shape,with a sharp central peak sitting on top of a broad “pedestal”�socle�. We consider this family of the gap solitons, whichhave no counterpart in the usual NLS model, as the mostimportant finding of the present work.

A generalized �perturbed� model that includes weakGVD in the nonlinear core was considered too, with a con-clusion that, in either case when the solitons are found in thesemi-infinite or finite gap in the unperturbed model, theysurvive �and remain stable� if the extra GVD is anomalous,and are destroyed if it is normal. The two-tier structure of thesolitons in the finite gap becomes more pronounced in thiscase. It was also shown that an attempt to set a soliton inmotion either leads to its relaxation back into a quiescent
state, or destroys the soliton.

ACKNOWLEDGMENTS

We thank the Guest Editors for their kind invitation tosubmit the paper to the special issue of Chaos. This workwas supported, in a part, by the Israel Science Foundationthrough Grant No. 8006/03.

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solitons in a linearly coupled system with separated dispersion and nonlinearity

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