Surface solitons in three dimensions

Q. E. Hoq,1 R. Carretero-González,2 P. G. Kevrekidis,3 B. A. Malomed,4 D. J. Frantzeskakis,5

Yu. V. Bludov,6 and V. V. Konotop6,7

1Department of Mathematics, Western New England College, Springfield, Massachusetts 01119, USA2Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, and Computational Science Research Center,

San Diego State University, San Diego, California 92182-7720, USA3Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA

4Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel5Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece

6Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Avenida Professor Gama Pinto 2,Lisboa 1649-003, Portugal

7Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8, Piso 6,Lisboa 1749-016, Portugal

�Received 5 June 2008; published 17 September 2008�

We study localized modes on the surface of a three-dimensional dynamical lattice. The stability of thesestructures on the surface is investigated and compared to that in the bulk of the lattice. Typically, the surfacemakes the stability region larger, an extreme example of that being the three-site “horseshoe”-shaped structure,which is always unstable in the bulk, while at the surface it is stable near the anticontinuum limit. We alsoexamine effects of the surface on lattice vortices. For the vortex placed parallel to the surface, the increasedstability-region feature is also observed, while the vortex cannot exist in a state normal to the surface. Moresophisticated localized dynamical structures, such as five-site horseshoes and pyramids, are also considered.

DOI: 10.1103/PhysRevE.78.036605 PACS number�s�: 05.45.Yv

I. INTRODUCTION

Surface waves have been a subject of interest in a varietyof contexts, including surface plasmons in conductors �1�and optical solitons in waveguide arrays �2�, surface wavesin isotropic magnetic gels �3�, water waves in the ocean ingeophysical hydrodynamics, and so on. Quite often, featuresexhibited by such wave modes have no analog in the corre-sponding bulk media, which makes their study especiallyrelevant. In particular, a great deal of interest has been drawnto nonlinear surface waves in optics. It was shown theoreti-cally �4� and observed experimentally �5� that discrete local-ized nonlinear waves can be supported at the edge of a semi-infinite array of nonlinear optical waveguide arrays. Suchsolitary waves were predicted to exist not only in self-focusing media �as in the above-mentioned works�, but alsobetween uniform and self-defocusing media �4,6�, or be-tween self-focusing and self-defocusing media �e.g., in �7��.They have been subsequently observed in media with qua-dratic �8� and photorefractive �9,10� nonlinearities. In thetwo-dimensional �2D� geometry, stable topological solitonshave been predicted in a saturable medium �11�, which con-stitute generalizations to lattice vortex solitons predicted inRef. �12�. Quasidiscrete vortex solitons have been experi-mentally observed in a self-focusing bulk photorefractivemedium �13�. Theoretical predictions for a variety of speciesof discrete 2D surface solitons �14–18� and corner modes�15,17�, as well as surface breathers �17�, have been reportedtoo. Subsequent work has resulted in the experimental obser-vation of 2D surface solitons, of both fundamental and mul-tipulse types, in photorefractive media �19�, as well as inasymmetric waveguide arrays written in fused silica �20�.Recently, surface solitons in more complex settings, such as

chirped optical lattices in one-dimensional �1D� and 2D situ-ations �21,22�, at interfaces between photonic crystals andmetamaterials �23�, and in the case of nonlocal nonlinearity�24,25�, have emerged.

Nearly all these efforts have been aimed at the study ofsurface solitons in 1D and 2D geometries. The only three-dimensional �3D� setting examined thus far assumed a trun-cated bundle of fiberlike waveguides, incorporating the tem-poral dynamics in longitudinal direction to produce 3D“surface light bullets” in Ref. �26� �the respective 2D surfacestructures were examined in Ref. �27��.

Our aim in the present work is to extend the analysis tosurface solitons in genuine 3D lattices. Our setup is relevantto a variety of applications including, e.g., crystals built ofmicroresonators trapping photons �28�, polaritons �29�, or,possibly, Bose-Einstein condensates in the vicinity of anedge of a strong 3D optical lattice �30,31�. In particular, wereport results for discrete solitons at the surface of a 3Dlattice, i.e., 3D localized states that are similar to relevantobjects studied in the 2D setting of Ref. �14�. Thus, we willstudy localized states such as dipoles and “horseshoes” abut-ting on a set of three lattice sites, but also states that arespecific to the 3D lattice. A variety of species of such soli-tons is examined below, and their stability on the surface iscompared to that in the bulk. Some localized structures, suchas dipoles, may be placed either normal or parallel to thesurface. We demonstrate that, typically, the enhanced contactwith the surface increases the stability region of the struc-ture. Physically, this conclusion may be understood by thefact that the surface reduces the local interactions to fewerneighbors, rendering the system “more discrete,” and hencemore stable �by pushing the medium further away from thecontinuum limit, where all solitons would be unstable againstthe collapse�. This effect is remarkable, e.g., for three-site

PHYSICAL REVIEW E 78, 036605 �2008�

1539-3755/2008/78�3�/036605�10� ©2008 The American Physical Society036605-1

horseshoes which are never stable in the bulk, but becomestabilized in the presence of the surface. However, the sur-face may also have an adverse effect, inhibiting the existenceof a particular mode. The latter trend is exemplified by dis-crete vortices, which, if placed parallel to the surface, featureenhanced stability as compared to the bulk-mode counter-part, but cannot exist with the orientation perpendicular tothe surface.

Surface-induced effects of a different kind, which are lessspecific to discrete systems, are induced by the interaction ofa particular localized mode with its fictitious “mirror image.”In terms of lattice models, the approach based on the analysisof the interaction of a real mode with its image was proposedin Ref. �32�.

To formulate the model, we introduce unit vectors e1= �1,0 ,0�, e2= �0,1 ,0�, and e3= �0,0 ,1� and define latticesites by n=� j=1

3 nje j with integer nj. We assume that the lat-tice occupies a semi-infinite space, n3�1, and its dynamicsobeys the discrete nonlinear Schrödinger �DNLS� equation inits usual form,

i�̇n + ���n + ���n�2�n = 0. �1�

Here �n is a complex discrete field, � is the coupling con-stant, �̇n stands for the time derivative, the parameter �= �1 determines the sign of the nonlinearity �focusing ordefocusing respectively�, and ��n is the 3D discrete Laplac-ian

��n � �j=1

3

��n+ej+ �n−ej

− 2s�n� �2�

for n3�2, while for n3=1 the term with subscript n−e3 is tobe dropped �note that e3 is the direction normal to the sur-face�.

It is interesting to point out here that an approach towardunderstanding the dynamics of Eq. �1� in the vicinity of thesurface can be based on the above-mentioned concept of thefictitious mirror image, formally extending the range of n3 upto n3=−�, by supplementing the equation with the antisym-metry condition,

�n1,n2,−n3� − �n1,n2n3

. �3�

Indeed, this condition implies �n1,n2,0�0, which is equiva-lent to the summation restriction in Eq. �2� as defined above.

To confine the analysis to localized solitary wave modes,we impose zero boundary conditions, �n→0 at n1,2→ ��and n3→�. Additionally, s= �1 in Eq. �2�—this parameteris introduced for convenience �see Sec. III B� and can befreely rescaled using the transformation �→�ei�t for an ap-propriate choice of � and time rescaling. Stationary solutionsto Eq. �1� will be sought for as �n=exp�it�un, where isthe frequency and the lattice field un obeys the equation

� − ��un�2�un − ��un = 0. �4�

Our presentation is structured as follows. The followingsection recapitulates the necessary background for the pre-diction of the existence and stability of lattice solitons. InSec. III, we report a bifurcation analysis for various surface

states, treated as functions of coupling constant �, with em-phasis on the comparison with bulk counterparts of thesestates. Section IV reports the study of the evolution of un-stable surface states. Finally, Sec. V summarizes our findingsand presents conclusions.

II. THEORETICAL BACKGROUND

First, we outline some general properties of the model.Equation �1� conserves two dynamical invariants, namely,the norm N,

N = �n3=1

n1,2=−�

�

��n�2, �5�

and the Hamiltonian H,

H = �n3=1

n1,2=−�

� ���j=1

3

��n*��n+ej

− s�n� + c.c.� +�

2��n�4 ,

�6�

where the asterisk stands for complex conjugation. Station-ary solutions to Eq. �4� with �= �1 are connected by thestaggering transformation �17,33�: if un is a solution for

some and �= +1, then �−1�n1+n2+n3un is a solution for ̃=12s− and �=−1. Consequently, it is sufficient to performthe analysis of stationary solutions, including their stability,for a single sign of the nonlinearity; thus, below we will fix�= +1 �corresponding to the case of on-site self-attraction�.

Solutions to Eq. �4� in the half space n3�1, subject to theboundary condition �n=0 for n3=0, as defined above, maybe continued antisymmetrically for the entire 3D space bysetting Un�un for n3�1 and Un�−un for n3−1. Then,according to results of Ref. �34�, this leads to an immediateconclusion, namely, that there exists a minimum norm Nminnecessary for the existence of localized surface states in thepresent model. In other words, no surface modes survive inthe limit of N→0. In this connection, it is relevant to notethat numerical findings that will be presented below wereobtained, of course, for finite cubic lattices where, strictlyspeaking, there is no lower limit for N necessary for theexistence of localized modes �17�. At this point, we have tospecify that in speaking about localized modes in a finitelattice we understand solutions that are localized on a num-ber of sites much smaller than the total number of sites in thechosen direction used for numerical simulations. Next werecall that, generally speaking, there exist several branches ofthe nonlinear localized modes, i.e., for given � one can findlocalized excitations at different values of the norm N. Usingthe natural terminology we refer to higher �lower� branchesin speaking about solutions with larger �smaller� norm. Inthis classification the surface modes we are dealing with cor-respond to higher branches of the solutions of the corre-sponding finite lattices, i.e., their norm cannot be made arbi-trarily small �see also the relevant discussion below in Sec.III B�.

To find solution families, we start with the anticontinuum�AC� limit �=0 �35�. In this limit, the lattice field is assumed

HOQ et al. PHYSICAL REVIEW E 78, 036605 �2008�

036605-2

to take nonzero values only at a few �“excited”� sites, whichdetermine the profile of the configuration to be seeded. Thecontinuation of the structure to ��0 is determined by theLyapunov reduction theorem �36�. More specifically, the so-lution is expanded as a power series in �, the solvabilitycondition at each order being that the corresponding projec-tion to the kernel generated by the previous order does notgive rise to secular terms �35�.

The linear stability is then studied, starting from the usualform of the perturbed solution,

�n = eit�un + �ane t + �bne *t� , �7�

where � is a formal small parameter, and is a stabilityeigenvalue associated with the eigenvector �= an ,b

n*�. Sub-

stituting this into Eq. �1� yields the linearized system

i an = − ��an + an − 2�un�2an − un2bn

*,

− i bn* = − ��bn

* + bn* − 2�un�2bn

* − un*2

an,

which can be cast in the form

�H�1,1� H�1,2�

H�2,1� H�2,2� �AB = i �A

B , �8�

where A and B are vectors composed of elements an and bn*,

respectively, while the matrices H�p,q� �p ,q� 1,2�� aregiven by

Hn,n��1,1� = − Hn,n�

�2,2� = �n,n�� + 6s� − 2�un��2�

− ��j=1

3

��n+ej,n� + �n−ej,n�� ,

Hn,n��1,2� = − Hn,n�

�2,1�* = − �n,n�un�2 . �9�

An underlying stationary solution is �spectrally� unstable ifthere exists a solution to Eq. �8� with Re� ��0. Otherwise,the stationary solution is classified as a spectrally stable one.As explained in Ref. �37�, the Jacobian of the above-mentioned solvability conditions is intimately connected tothe full eigenvalue problem. More specifically, if the eigen-values � of the M �M eigenvalue problem of the Jacobian�where M is the number of excited sites at the AC limit�, thenthe near-zero eigenvalues of the full stability problem can bepredicted to be =�2��p/2, where p is the number of latticesites that separate the adjacent excited nodes of the configu-ration at the AC limit.

III. BIFURCATION ANALYSIS

A. Existence and stability of surface structures

In this section we study, by means of numerical methods,the existence and stability of various 3D configurations, andcompare the results to the corresponding analytical predic-tions. These configurations are obtained by starting from theAC limit ��=0�, and are continued to ��0, using fixed-pointiterations. For all the numerical results presented in this

work, we fix the normalization =1 �see Eq. �4��, and use alattice of size 13�13�13, unless stated otherwise. Also, forthe presentation of the numerical results, we replace the trip-let �n1 ,n2 ,n3� by �l ,n ,m�, i.e., the surface corresponds tom=1.

We start by examining dipoles aligned parallel or normalto the surface. Figure 1�a� shows the norm of such statesversus the coupling constant �, while Fig. 1�b� depicts theimaginary part of the stability eigenvalues for the bulk di-pole, produced by the theory outlined in the previous section�dashed �black� lines�, and by the numerical computations�solid �blue� lines�. It is worth mentioning that, for all thedifferent configurations that we report in this paper, we dis-

0 0.05 0.1 0.15 0.22345

ε

N

0 0.05 0.1 0.15 0.20

0.5

1

ε

λ i

(a)

(b)

0 0.05 0.1 0.15 0.20

0.05

0.1

ε

Max

(λr)

0.112 0.114 0.116 0.118 0.12 0.1220

0.02

0.04

ε

Max

(λr)

(c)

(d)

1 2 3468

6

7

8

mn

l

−0.01 0 0.01

−2

0

2

λr

λ i

(e)(f)

1 2 3686

7

8

mn

l

−5 0 5

x 10−3

−2

0

2

λr

λ i

(g) (h)

FIG. 1. �Color online� Results for the dipoles oriented paralleland normal to the surface. �a� Norm N versus the lattice couplingconstant �. �b� Imaginary part of the linear stability eigenvalue:solid �blue� and dashed �black� lines correspond, respectively, tonumerically found and analytically predicted forms. �c� Real part ofthe critical �in�stability eigenvalue: the dashed �red� and solid �blue�lines depict the normal- and parallel-oriented dipoles, respectively,while the dashed-dotted �green� line corresponds to the bulk dipole.�d� �In�stability eigenvalue for the parallel surface dipole placed atdistances from the surface starting from zero and up to five latticeperiods away �curves right to left�. �e�, �g� Configurations and �f�,�h� corresponding spectral stability planes just above the instabilitythreshold. The level contours, corresponding to Re�ul,n,m�= �0.5 maxul,n,m� are shown, respectively, in dark gray �blue� andgray �red�. The instability thresholds for the dipoles oriented paral-lel and normal to the surface are, respectively, �=0.117 and 0.120.For comparison, the threshold for the bulk dipole is �=0.114.

SURFACE SOLITONS IN THREE DIMENSIONS PHYSICAL REVIEW E 78, 036605 �2008�

036605-3

play the imaginary part of the stability eigenvalue only forthe bulk mode since the difference between the curves for thedifferent variants �bulk, parallel, or normal to the surface� isminimal. It should be noted, however, that the contact withthe surface may produce higher-order �smaller� eigenvaluesthat are not present in its bulk counterpart �results not shownhere�. The theoretical prediction for the stability eigenvaluesis = �2��i, which, as expected, is the same as in the out-of-phase �twisted� 1D mode analyzed in Ref. �37�, since thestructure is nearly one-dimensional, along the line connect-ing the two excited sites. Figure 1�c� compares the largestinstability growth rate as a function of � for the bulk dipoles�dashed-dotted �green� line� and those oriented normally andparallel to the surface �dashed �red� and solid �blue� lines,respectively�. It is seen that the stability interval of the di-poles increases as its contact with the surface strengthens, inaccordance with the arguments presented above. In the caseof the bulk dipole, the instability occurs for values of thecoupling constant between �0=0.114 and �1=0.115. Fromnow on, when reporting computed instability thresholds, wewill use the lower bound for � �e.g., �0 in the above example�with the understanding that we always used the same reso-lution in �. For the dipole set normally to the surface, weobserve the onset of instability at �=0.117, while for theparallel-oriented one at �=0.120. In Figs. 1�e�–1�h� we alsodepict the shapes of the normal and parallel dipoles, justbelow the instability threshold, along with their correspond-ing spectral stability planes.

The stabilizing effects exerted by the surface depend, in agreat measure, on the distance of the configuration from the

surface, namely, the further away the configuration from thesurface, the smaller the effect is. This property is clearly seenin Fig. 1�d�, where we plot the �in�stability eigenvalue as afunction of the coupling for several values of the separationof the parallel dipole from the surface. The curves, from rightto left, depict the results for the dipole set at a distance of0,1,…,5 sites away from the surface �the 0 site refers to thedipole sitting on the surface�. As the panel demonstrates, thestability interval is reduced as the dipole is pulled away fromthe surface, converging toward a bulk dipole.

Let us now consider the horseshoe configurations, forwhich the presence of the surface is critical to their stability.In Fig. 2 we depict the properties of a three-site horseshoe,which actually is a truncated version of a quadrupole �cf. the2D situation �14��. As before, Fig. 2�a� shows the norm ver-sus �, while Figs. 2�b� and 2�c� compare the stability of thebulk horseshoe �the dashed-dotted line� and ones built nearthe surface �the solid line�. While the bulk horseshoes arealways unstable, similar to their 2D counterparts �14�, theones placed near the surface are stable at small �, destabiliz-ing at �=0.239. Figures 2�d� and 2�e� show the configurationfor the coupling just below the instability threshold, alongwith the corresponding spectral plane. The analytical expres-sions for stable eigenvalues are =0, = �2�3�i, =O��2� �cf. the expressions obtained in Ref. �14� for the 2Dhorseshoes�.

Figure 3 illustrates the same features as before, but for thefive-site horseshoe. Unlike its three-site cousin, the bulkfive-site horseshoe is stable up to a critical value of the cou-pling, �=0.205, while the surface variant has it stability re-gion ��0.211. The eigenvalues of the linearization in thiscase can be computed similarly to those for the three-sitemodes �14�, as outlined above �cf. also Ref. �35��, which

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

ε

λ i

0 0.1 0.2 0.3 0.4

5

10

ε

N(a)

(b)

0 0.1 0.2 0.3 0.40

0.2

ε

Max

(λr) (c)

1 2 34686

7

8

mn

l

−0.02 0 0.02−5

0

5

λr

λ i

(d)(e)

FIG. 2. �Color online� Stability of the three-site horseshoe. Pan-els are similar to those in Fig. 1. �c� compares the critical stabilityeigenvalue, as a function of the lattice coupling �, for the surfaceand bulk horseshoes �solid �blue� and dashed-dotted �green� lines,respectively�. The bulk horseshoe is always unstable �due to apurely real, higher-order eigenvalue�, while the corresponding sur-face configurations have a stability region �the corresponding eigen-value becomes imaginary in this case�. �d� and �e� correspond to thesurface horseshoe just above the stability threshold of �=0.239.

0 0.05 0.1 0.15 0.2 0.25 0.35

10

15

ε

N

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

ε

λ i

(a)

(b)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

ε

Max

(λr) (c)

1 2 3 45

1067

8

mn

l

−0.02 0 0.02−4−2024

λr

λ i

(d)(e)

FIG. 3. �Color online� Stability for the five-site horseshoe at thesurface. Panels are identical to those in Fig. 2. In this case, thestability threshold is at �=0.211, while for the bulk five-site horse-shoe it is �=0.205. �d� and �e� depict the configuration and thecorresponding linear stability spectrum just above the critical pointof �=0.211.

HOQ et al. PHYSICAL REVIEW E 78, 036605 �2008�

036605-4

eventually yields =3.8042�i, =2.8284�i, =2.3511�i, =O��2�, and =0, in good agreement with the correspondingnumerical results, as shown in Fig. 3�b�.

Next we consider the quadrupole configuration �see Fig.4�. The surface again exerts a stabilizing effect, albeit aweaker one, when the quadrupole is placed normally andparallel to the surface. In the bulk, the quadrupole loses sta-bility at �=0.068, while the normal and parallel surface qua-drupoles have stability thresholds, respectively, at �=0.070and 0.071. The analytical approximation for the stability ei-genvalues in this case are =�8�i �a double eigenvalue�, =2��i, and a zero eigenvalue, which accurately capture thenumerical findings depicted in Fig. 4�b�.

In Fig. 5 we present the results for four-site vortices. Thisconfiguration, in contrast to the previous ones, is describedby a complex solution. In the AC limit, the vortex occupiesthe same excited sites as the above-mentioned quadrupole,but the phase profile, 0,� /2,� ,3� /2�, emulates that of thevortex of charge 1 �12,35�. The bulk four-site vortex �whichwas discussed in Ref. �38�� loses its stability at �=0.438,while the vortex parallel to the surface features an extendedstability region, up to �=0.505. However, the surface in thiscase prohibits the existence of a vortex that would be ori-ented normally to the surface layer, similarly to what wasfound for 2D lattice vortices �14�.

The simplest explanation for the complete absence of thesolution normal to the surface, compared with that of anexisting vortex waveform set parallel to the surface can ar-guably be traced in the interaction of such vortices in the half

space with their fictitious image �if the domain is equiva-lently extended to the full space�. In the case of the vortexparallel to the surface, the situation is tantamount to the vor-tex cube structures examined in �39,40�, for which it wasestablished in �40� that the persistence conditions are satis-fied �and, in fact, that such structures consisting of two out-of-phase vortices should be linearly stable close to the AClimit�. On the other hand, for the case normal to the surface,by examining the relevant interactions it can be observed �atan appropriately high order� that the persistence conditionsof �35,37,40� cannot be satisfied and hence the structure can-not be continued beyond the AC limit. That is why the struc-ture can never be observed to exist, irrespective of the small-ness of �.

The next species of stationary lattice solutions is apyramid-shaped structure, with characteristics displayed inFig. 6, whose base is a rhombus composed of four sites. Theremaining out-of-plane vertex site must have phase 0 or �,since the phase values � /2 and 3� /2 at this site do notproduce a solution. The full set of pyramids �bulk, normal,parallel—see Figs. 6�d�–6�f�� is completely unstable, as seenin Fig. 6�c�, the surface producing no stabilizing effect on it.This strong instability actually arises at the lowest order inthe analytical eigenvalue calculations, which yield =2�5�i, =2�2�i, =2�, =0, and =O��2�, once againin very good agreement with the full numerical results ofFig. 6.

0 0.02 0.04 0.06 0.08 0.14

6

8

ε

N

0 0.02 0.04 0.06 0.08 0.10

0.5

1

ε

λ i(a)

(b)

0.065 0.07 0.075 0.080

0.02

0.04

ε

Max

(λr) (c)

1 2 35678

5678

mn

l

−2 0 2x 10−3

−2

0

2

λr

λ i

(d)(e)

FIG. 4. �Color online� Stability of quadrupole modes. The lay-out is similar to that in Fig. 3. In �c�, due to the close proximity ofthe thresholds, a close-up is shown for the critical stability eigen-value versus the lattice coupling constant � for the parallel andnormal surface modes, and the bulk one �solid �blue� and dashed�red� lines, and the dashed-dotted �green� line, respectively�. Thethreshold for the bulk mode is �=0.068, while for the normal andparallel quadrupoles it is, respectively, �=0.070 and 0.071. As be-fore, �d� and �e� show the configuration just above the instabilitythreshold along with its corresponding spectral-stability plane.

0 0.2 0.4 0.6 0.8

10

20

ε

N

(a)

0 0.2 0.4 0.6 0.80

0.5

1

ε

λ i

(b)

0 0.2 0.4 0.6 0.80

0.5

ε

Max

(λr) (c)

1 2 3456

4567

mn

l

−0.01 0 0.01−10

0

10

λr

λ i

(d)(e)

FIG. 5. �Color online� Stability of the four-site vortex in a gridof size 11�11�11. The dashed-dotted and solid lines show thebulk vortex and the one parallel to the surface, respectively. Thelayout is similar to that of the above figures. Instability in the bulkoccurs at �=0.438, and in the parallel surface vortex at �=0.505.The vortex cannot exist with the orientation normal to the surface.Panels �d� and �e� show the parallel surface vortex just above theinstability threshold of �=0.485. As in the previous figures, thelevel contours corresponding to Re�ul,n,m�= �0.5 maxul,n,m� areshown, respectively, in dark gray �blue� and gray �red�, while thecomplementary level contours, defined as Im�ul,n,m�= �0.5 maxul,n,m�, are shown by light gray �green� and very lightgray �yellow� hues, respectively.

SURFACE SOLITONS IN THREE DIMENSIONS PHYSICAL REVIEW E 78, 036605 �2008�

036605-5

B. Small-amplitude modes in a finite lattice

Since our numerical investigation of the surface modesuses a finite lattice, which allows the existence of small-amplitude modes �ones with the zero threshold in terms ofthe norm—see the discussion in Sec. II�, here we brieflyconsider the modes in a finite lattice having the small-amplitude limit. Our aim is to show that these modes belongto lower branches, as compared with the “normal” surfacemodes considered above. To this end, we concentrate on thelattice of size M �M �M lattice, subject to zero boundaryconditions, which imply that the discrete Laplacian �2� ismodified at surfaces nj =1 and nj =M �j=1, . . . ,3� by settingthe fields at sites n−e j and n+e j, respectively, equal to zero.For the sake of definiteness, we fix here s=−1 in Eq. �2�.

To determine the norm N of small-amplitude modes wefollow Ref. �17�, and look for a solution to Eq. �4� with theamplitude un and coupling constant � being represented asseries,

un = �u0,n + �2u2,n + O��3� ,

� = �0 + �2�2 + O��3� , �10�

in powers of the small parameter ���8N / �M +1�3�1,which vanishes in the limit of the infinite lattice �M→��; inother words, small � characterizes the “largeness” of the lat-tice. We focus on real solutions here.

Substituting expansions �10� into Eq. �4� and gatheringterms of the same order in �, we rewrite Eq. �4� in the formof a set of equations:

uj,n − �0�uj,n = Fj,n. �11�

Here F0,n=0, F2,n=��2 /�0�u0,n+ �u0,n�3; hence Eq. �11�with j=0 gives rise to a linear eigenmode,

u0,n�m� =

j=1

3

sin��njmj

M + 1 , �12�

with the corresponding approximation for the lattice cou-pling constant,

�0�m� = �6 + 2�

j=1

3

cos� �mj

M + 1�−1

, �13�

parametrized by the vector m= �m1 ,m2 ,m3�. At the sametime, considering the solvability conditions for j=2, whichamounts to demanding the orthogonality of F2,n and u0,n, weobtain corrections to the coupling constants,

� = �0�m� −

�2�0�m�

64 j=1

3

�3 + �mj,�M+1�/2� . �14�

It follows from Eq. �14� that each of the linear modes �12� isuniquely extended into a small-amplitude nonlinear one.These modes are characterized by the linear dependence ofthe norm on the coupling constant �:

N�m� =8�M + 1�3��0

�m� − ��

�0�m�

j=1

3

�3 + �mj,�M+1�/2�

. �15�

From Eq. �15� it follows that each mode, parametrized byvector m, exists when � belongs to the interval 0��0

�m�.The validity of approximation �15� is corroborated by thecoincidence of analytical and numerical results in the vicin-ity of �0

�m� �as shown in Fig. 7�, where these modes reachtheir small-amplitude limit. Such a property of these modes

0 0.05 0.1 0.15 0.2 0.25 0.35

10

ε

N(a)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

ε

λi

(b)

0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

ε

Max

(λr) (c)

6 7 8678

678

mn

l

1 2 3 4678

678

mn

l

1 2 3 4678

678

mn

l

(d) (e) (f)

FIG. 6. �Color online� Instability of pyramid-shaped structures.This configuration abuts on the base in the form of a rhombus, andincludes the out-of-plane site with zero phase. Three variants of thisconfiguration are displayed in �d�–�f�: bulk and normal and parallelto the surface, respectively. The stability of the three different vari-ants of the pyramid is essentially identical, all three of them beingunstable.

ε

N

0 0.02

(a)

0.04 0.06 0.080

5

10

15

m=(1,1,1)m=(1,1,2)

m=(1,2,2)

ε

max

(λr)

0 0.02 0.04 0.06 0.080

0.2

(b)0.4

m=(1,1,1)m=(1,1,2)

m=(1,2,2)

FIG. 7. �Color online� Low-amplitude modes in a finite grid ofsize 9�9�9 with =1.0. �a� Norm N versus coupling constant �for several modes whose low-amplitude limit is parametrized byvectors m, as calculated numerically and predicted by approxima-tion �15� �solid and dashed lines, respectively�. For comparison, thedashed-dotted line depicts the norm for surface normal dipole. �b�Real part of the critical �in�stability eigenvalue, calculatednumerically.

HOQ et al. PHYSICAL REVIEW E 78, 036605 �2008�

036605-6

differs considerably from the case of the surface modes,which do not possess a small-amplitude limit and requiresome minimal value of the norm �for the normal dipole, de-picted in Fig. 7 by the dashed-dotted line, the minimal normis �1.262�. Figure 7�b� shows that only the mode param-etrized by vector m= �1,1 ,1� is stable for � close to �0

�m�,while other modes are completely unstable.

IV. DYNAMICS

In this section we examine the nonlinear evolution of thevarious configurations, displaying the results in a set of fig-ures �see Figs. 8–12�. In each case, the evolution is initiatedat a value of the coupling � taken beyond the instability

threshold, and an initial small random perturbation is appliedin order to expedite the onset of the instability.

All the figures display the evolution of the instability atsix different moments of time, starting at t=0, and ending ata time well beyond the point at which the instability mani-

67

86

7

7l(a) t=0

67

86

7

7l

t=29

67

86

7

7l

t=30

67

86

7

7

mn

l

t=38

67

86

7

7

mn

lt=47

67

86

7

7

mn

l

t=50

12

36

7

7l

(b) t=0

12

36

7

7l

t=43

12

36

7

7l

t=48

12

36

7

7

mn

l

t=49

12

36

7

7

mn

l

t=51

12

36

7

7

mn

l

t=100

12

36

78

7l

(c) t=0

12

36

78

7l

t=39

12

36

78

7l

t=48

12

36

78

7

mn

l

t=51

12

36

78

7

mn

l

t=53

12

36

78

7

mn

l

t=100

FIG. 8. �Color online� Evolution of unstable dipoles: �a� a bulkdipole; �b� and �c� dipoles placed parallel and normal to the surface,respectively. In all cases, the dipole is subject to an oscillatoryinstability, which is responsible for the eventual concentration ofmost of the norm at a single site �i.e., the transition to a monopole�.Parameters are =1, �=0.2, the lattice has a size of 13�13�13,and times are indicated in the panels. All isocontour plots are de-fined as Re�ul,n,m�= �0.75=Im�ul,n,m�, and the initial configurationswere perturbed with random noise of amplitude 0.01. The codingfor the isocontours is as follows: dark gray �blue� and gray �red�colors pertain to isocontours of the real part of the solutions, whilethe light gray �green� and very light gray �yellow� colors correspondto the isocontours of the imaginary part.

6 7 8 9678

678

(a) t=0

l

6 7 8 9678

678t=22

l

6 7 8 9678

678t=25

l

6 7 8 9678

678

mn

t=26

l

6 7 8 9678

678

mn

t=29.5

l

6 7 8 9678

678

mn

t=50

l

1 2 3 4678

678

(b) t=0

l

1 2 3 4678

678t=27

l

1 2 3 4678

678t=35

l

1 2 3 4678

678

mn

t=43

l

1 2 3 4678

678

mn

t=63

l

1 2 3 4678

678

mn

t=100

l

FIG. 9. �Color online� Evolution of the unstable three-site horse-shoes: �a� bulk three-site horseshoe and �b� horseshoe oriented nor-mally to the surface. In both cases, the unstable horseshoe is subjectto an oscillatory instability, which leads to the eventual concentra-tion of most of the norm in a single-site structure. The isocontoursand parameters are the same as in Fig. 8 except that �=0.3.

686

8

678

(a) t=0

l

686

8

678t=20

l

686

8

678t=23

l

686

8

678

mn

t=24

l

686

8

678

mn

t=26

l

686

8

678

mn

t=50

l

246

8

678

(b) t=0

l

246

8

678t=17

l

246

8

678t=18

l

246

8

678

mn

t=23

l

246

8

678

mn

t=24

l

246

8

678

mn

t=50

l

FIG. 10. �Color online� Evolution of unstable five-site horse-shoes: �a� the bulk horseshoe, and �b� the five-site horseshoe ori-ented normal to the surface. In both cases, the unstable horseshoe issubject to an oscillatory instability, which triggers the transition to amonopole. The isocontours and parameters are the same as inFig. 9.

SURFACE SOLITONS IN THREE DIMENSIONS PHYSICAL REVIEW E 78, 036605 �2008�

036605-7

fests itself. All configurations that were predicted above to beunstable through nonzero real parts of the �in�stability eigen-value indeed exhibit instability dynamics, which eventu-ally results in a transition to a different configuration. In thecase of the dipoles and horseshoes, Figs. 8–10 show a spon-taneous transition to monopole patterns, i.e., those centeredaround a single excited site. On the other hand, in the case ofthe vortices and pyramids shown in Figs. 11 and 12, a fewsites may remain essentially excited at the end of the evolu-tion. The monopole is, obviously, the most robust dynamicalstate in the lattice system, with the widest stability interval,in comparison with other discrete structures. This simpleststate becomes unstable, for given , only at values of thecoupling constant �� �38�. Another structure with a rela-tively wide stability region is the dipole �the more stable thewider the distance between its constituent sites �39��, conso-nant with the observation that some of the structures �espe-cially ones with a large number of excited sites, such asvortices and pyramids� dynamically transform into dipoles.

Generally speaking, the exact scenario of the nonlinearevolution and the finally established state depend on detailsof the initial perturbation. In the figures, each configurationis shown by isolevel contours of distinct hues. In particular,dark gray �blue� and gray �red� are isocontours of the realpart of the solutions, while the light gray �green� and verylight gray �yellow� colors depict the imaginary part of thesame solutions.

A case that needs further consideration is that of the three-site horseshoe. As observed from the stability analysis pre-sented in Fig. 2, this horseshoe in the bulk gives rise to asmall unstable purely real eigenvalue for all values of �; see

the lower green dashed-dotted curve in Fig. 2�c� of the fig-ure. Despite the presence of this eigenvalue, the evolution ofthe unstable bulk three-site horseshoe is predominantlydriven by the unstable complex eigenvalues, if any �in fact,for ��0.226; see the dashed-dotted �green� line of Fig. 2�c��.A careful analysis of the instability corresponding to thesmall purely real eigenvalue for ��0.226 �i.e., before thecomplex eigenvalues become unstable� reveals that the cor-responding dynamics amounts to an extremely weak ex-change of the norm between the two in-phase excited sites�see Fig. 2�. The norm exchange is driven by the correspond-ing unstable eigenfunction, which looks like a dipole posi-tioned at the two aforementioned in-phase sites. The diffi-culty in observing this evolution mode is explained by thefact that, in the course of the norm exchange, only �0.1% ofthe total norm is actually transferred between the two sites.Furthermore, as mentioned earlier, the corresponding smallreal eigenvalue is completely suppressed by the surface �seeFig. 2�c��. It is worth noting that such stable three-site horse-shoe surface structures may also be generated by the evolu-tion of more complex unstable wave forms, such as the five-

67

86

7

6

7

l(a) t=0

67

86

7

6

7

l

t=24

67

86

7

6

7

l

t=27

67

86

7

6

7

mn

l

t=31

67

86

7

6

7

mn

lt=47

67

86

7

6

7

mn

l

t=100

1 2 36

7

6

7

(b) t=0

l

1 2 36

7

6

7

t=42

l

1 2 36

7

6

7

t=43l

1 2 36

7

6

7

mn

t=44

l

1 2 36

7

6

7

mn

t=46

l

1 2 36

7

6

7

mn

t=100

l

FIG. 11. �Color online� Evolution of unstable vortices: �a� thebulk vortex for �=0.3 and �b� the vortex parallel to the surface, for�=0.6 and =1. The isocontour plots are defined by Re�ul,n,m�= �1=Im�ul,n,m�.

6 7 86785

6

7

l

(a) t=0

6 7 86785

6

7

l

t=16

6 7 86785

6

7

l

t=96

6 7 86785

6

7

mn

l

t=156

6 7 86785

6

7

mn

l

t=188

6 7 86785

6

7

mn

l

t=268

1 2 36785

6

7

l

(b) t=0

1 2 36785

6

7

l

t=20

1 2 36785

6

7

l

t=66

1 2 36785

6

7

mn

l

t=82

1 2 36785

6

7

mn

l

t=182

1 2 36785

6

7

mn

l

t=400

1 2 36786

7

8l

(c) t=0

1 2 36786

7

8

l

t=30

1 2 36786

7

8

l

t=156

1 2 36786

7

8

mn

l

t=212

1 2 36786

7

8

mn

l

t=222

1 2 36786

7

8

mn

l

t=400

FIG. 12. �Color online� Evolution of unstable pyramids. Panels�a�, �b�, and �c� display, respectively, the transformation of a bulkpyramid, and of pyramids oriented normal and parallel to the sur-face, for �=0.2.

HOQ et al. PHYSICAL REVIEW E 78, 036605 �2008�

036605-8

site pyramids placed normally to the surface; see the bottompanel in Fig. 12.

V. CONCLUSIONS

In this work, we have investigated localized modes in thevicinity of a two-dimensional surface, in the framework ofthe three-dimensional DNLS equation, which is a prototypi-cal model of nonlinear dynamical lattices. We have foundthat the surface may readily stabilize localized structures thatare unstable in the bulk �such as three-site horseshoes�, and,on the other hand, it may inhibit the formation of some otherstructures that exist in the bulk �such as vortices that areoriented normal to the surface, although ones parallel to thesurface do exist and have their stability region; a qualitativeexplanation for these features was proposed, based on theanalysis of the interaction of the vortical state with its mirrorimage�. The most typical surface-induced effect is the expan-sion of the stability intervals for various solutions that exist

in the bulk and survive in the presence of the surface. Thisfeature may be attributed to the decrease, near the surface, ofthe number of neighbors to which excited sites couple, sincethe approach to the continuum limit, i.e., the strengthening ofthe linear couplings to the nearest neighbors, is responsiblefor the onset of the instability or disappearance of all thelocalized stationary states in the three-dimensional dynami-cal lattice.

On the other hand, while the techniques elaborated inRefs. �35,37,40� for the analysis of localized states in bulklattices are quite useful in understanding the dominant stabil-ity properties of the solutions, the surface gives rise to spe-cific effects, such as the stabilization of higher-order solu-tions or the suppression of some types of vortex structures,which cannot be explained by these methods. Therefore, itwould be very relevant to modify these techniques, whichare based on the Lyapunov-Schmidt reductions, so as to takethe presence of the surface into consideration.

�1� W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature �London�424, 824 �2003�.

�2� D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature�London� 424, 817 �2003�.

�3� S. Bohlius, H. R. Brand, and H. Pleiner, Z. Phys. Chem. 220,97 �2006�.

�4� K. G. Makris, S. Suntsov, D. N. Christodoulides, and G. I.Stegeman, Opt. Lett. 30, 2466 �2005�; M. I. Molina, R. A.Vicencio, and Yu. S. Kivshar, ibid. 31, 1693 �2006�.

�5� S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stege-man, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M.Sorel, Phys. Rev. Lett. 96, 063901 �2006�.

�6� Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Phys. Rev.Lett. 96, 073901 �2006�.

�7� D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis,A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, Phys. Rev. E74, 036602 �2006�.

�8� G. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N.Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, Opt.Express 14, 5508 �2006�.

�9� C. R. Rosberg, D. N. Neshev, W. Królikowski, A. Mitchell, R.A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. Lett.97, 083901 �2006�.

�10� E. Smirnov, M. Stepić, C. E. Rüter, D. Kip, and V. Shandarov,Opt. Lett. 31, 2338 �2006�.

�11� Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner,Opt. Express 14, 4049 �2006�.

�12� B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601�2001�.

�13� D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S.Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev.Lett. 92, 123903 �2004�.

�14� H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, Phys. Rev. E 75, 056605�2007�.

�15� K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stege-

man, O. Manela, and M. Segev, Opt. Lett. 31, 2274 �2006�.�16� Y. V. Kartashov and L. Torner, Opt. Lett. 31, 2172 �2006�.�17� Yu. V. Bludov and V. V. Konotop, Phys. Rev. E 76, 046604

�2007�.�18� R. A. Vicencio, S. Flach, M. I. Molina, and Yu. S. Kivshar,

Phys. Lett. A 364, 274 �2007�.�19� X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N.

Christodoulides, and G. I. Stegeman, Phys. Rev. Lett. 98,123903 �2007�.

�20� A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte,A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903�2007�.

�21� Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A76, 013831 �2007�.

�22� M. I. Molina, Y. V. Kartashov, L. Torner, and Yu. S. Kivshar,Phys. Rev. A 77, 053813 �2008�.

�23� A. Namdar, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. A75, 053812 �2007�.

�24� B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N.Christodoulides, Phys. Rev. Lett. 98, 213901 �2007�.

�25� F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 033829�2008�.

�26� D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, Opt.Lett. 32, 3173 �2007�.

�27� D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, Opt.Lett. 32, 2091 �2007�.

�28� J. E. Heebner and R. W. Boyd, J. Mod. Opt. 49, 2629 �2002�;P. Chak, J. E. Sipe, and S. Pereira, Opt. Lett. 28, 1966 �2003�.

�29� J. J. Baumberg, P. G. Savvidis, R. M. Stevenson, A. I. Tartak-ovskii, M. S. Skolnick, D. M. Whittaker, and J. S. Roberts,Phys. Rev. B 62, R16247 �2000�; P. G. Savvidis and P. G.Lagoudakis, Semicond. Sci. Technol. 18, S311 �2003�.

�30� O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179�2006�.

�31� I. Bloch, Nat. Phys. 1, 23 �2005�.�32� See the recent work of K. G. Makris and D. N. Christodoul-

SURFACE SOLITONS IN THREE DIMENSIONS PHYSICAL REVIEW E 78, 036605 �2008�

036605-9

ides, Phys. Rev. E 73, 036616 �2006�, as well as earlier worksuch as K. Ø. Rasmussen, D. Cai, A. R. Bishop, and N.Grønbech-Jensen, ibid. 55, 6151 �1997� and P. G. Kevrekidis,I. G. Kevrekidis, and B. A. Malomed, J. Phys. A 35, 267�2002�.

�33� G. L. Alfimov, V. A. Brazhnyi, and V. V. Konotop, Physica D194, 127 �2004�.

�34� M. I. Weinstein, Nonlinearity 12, 673 �1999�.�35� D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis,

Physica D 212, 20 �2005�.�36� M. Golubitsky, D. G. Schaeffer, and I. Stewert, Singularities

and Groups in Bifurcation Theory �Springer, New York, 1985�,Vol. 1.

�37� D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis,Physica D 212, 1 �2005�.

�38� P. G. Kevrekidis, B. A. Malomed, D. J. Frantzeskakis, and R.Carretero-González, Phys. Rev. Lett. 93, 080403 �2004�.

�39� R. Carretero-González, P. G. Kevrekidis, B. A. Malomed, andD. J. Frantzeskakis, Phys. Rev. Lett. 94, 203901 �2005�.

�40� M. Lukas, D. Pelinovsky, and P. G. Kevrekidis, Physica D237, 339 �2008�.

HOQ et al. PHYSICAL REVIEW E 78, 036605 �2008�

036605-10