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arXiv:0812.1301v1 [nlin.PS] 6 Dec 2008

Interlaced solitons and vortices in coupled DNLS lattices

J. Cuevas,1Q.E. Hoq,2H. Susanto,3and P.G. Kevrekidis4

1Grupo de F´ ısica No Lineal, Universidad de Sevilla. Departamento de F´ ısica Aplicada

I. Escuela Universitaria Polit´ ecnica, C/ Virgen de´Africa, 7, E-41011 Sevilla, Spain

2Department of Mathematics, Western New England College, Springfield, Massachusetts 01119, USA

3School of Mathematical Sciences, University of Nottingham,

University Park, Nottingham, NG7 2RD, United Kingdom

4Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA

In the present work, we propose a new set of coherent structures that arise in nonlinear dynamical

lattices with more than one components, namely interlaced solitons. These are waveforms in which

in the relevant anti-continuum limit, i.e. when the sites are uncoupled, one component has support

where the other component does not. We illustrate systematically how one can combine dynamically

stable unary patterns to create ones such for the binary case of two-components.

dimensional setting, we provide also a detailed theoretical analysis of the existence and stability

of these waveforms, while in higher dimensions, where such analytical computations are far more

involved, we resort to corresponding numerical computations. Lastly, we perform direct numerical

simulations to showcase how these structures break up, when exponentially or oscillatorily unstable,

to structures with a smaller number of participating sites.

In the one-

PACS numbers:

INTRODUCTION

One of the highly active areas of investigation of Hamiltonian nonlinear systems over the past decade has been

the examination of nonlinear dynamical lattices of the discrete nonlinear Schr¨ odinger (DNLS) type. Chiefly, this

development has arisen due to the multitude of applications of pertinent models that have emerged in areas such as

nonlinear optics and atomic physics.

More specifically, in the optical context, the setting of fabricated AlGaAs waveguide arrays [1] has been one of

the most prototypical ones for the application of DNLS models. There, the interplay of discreteness and nonlinearity

revealed many interesting features including Peierls-Nabarropotential barriers, diffraction and diffraction management

[2], and gap solitons [3], among others; see also the reviews [4, 5] and references therein.

Another recent development, which also promoted the analysis of discrete systems in connection with nonlinear

optics was the proposal [6] and creation [7, 8] of optically induced photonic lattices in photorefractive crystals such

as SBN. This paved the way for the observation of a large set of exciting nonlinear wave related phenomena in such

crystals. As a representative subset, we mention here the formation of patterns such as dipole [9], quadrupole [10]

and necklace [11] solitary waves, impurity modes [12], discrete vortices [13, 14], rotary waves [15], higher order Bloch

modes [16] and gap vortices [17], two-dimensional (2D) Bloch oscillations and Landau-Zener tunneling [18], wave

formation in honeycomb [19], hexagonal [20] and quasi-crystalline lattices [21], and recently the study of Anderson

localization in disordered photonic lattices [22]. Although this setting is mostly studied in the continuum context with

a periodic potential (and sometimes in the presence of the inherent crystal anisotropy), it has also spurred a number

of studies in the DNLS context with the saturable photorefractive nonlinearity [23, 24].

Lastly, another physical realization of such nonlinear dynamical lattices arose over the past few years in atomic

physics through the examination of Bose-Einstein condensates (BECs) trapped in periodic potentials. There, once

again, a reduction of the relevant model can be formulated in the tight-binding approximation within the mean-field

limit, reducing the so-called Gross-Pitaevskii equation with a periodic potential to a genuinely discrete nonlinear

Schr¨ odinger equation [25].

In both the nonlinear optical and in the atomic physics setting discussed above, multi-component systems were

also examined in recent investigations. More specifically, the first observations of discrete vector solitons in optical

waveguide arrays were reported in [26], the emergence of multipole patterns in vector photorefractive crystals was

presented in [27], while numerous experiments with BECs were directed towards studies of mixtures of different spin

states of87Rb [28, 29] or23Na [30] and even ones of different atomic species such as41K–87Rb [31] and7Li–133Cs

[32]. It should be noted that while the above BEC experiments did not include the presence of an optical lattice, the

addition of such an external optical potential is certainly feasible within the present experimental capabilities [33].

Our aim in the present work is to propose and analyze a family of solutions particular to multicomponent (in

particular, binary, although more-component generalizations are certainly possible) systems of DNLS equations. We

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dub these proposed solutions “interlaced” discrete solitons and vortices, a name stemming from the feature that the

profiles of the modes in the two interacting components will have a vanishing intersection of excited sites in the

extreme discrete limit of zero coupling between adjacent nodes of the lattice. In these structures, the first component

will be excited where the second component is not and vice-versa. In the one-dimensional case, we show how to

interlace in a stable fashion simple, as well as more elaborate, bound states of the system [34]. For such solutions, we

consider their existence and stability properties also from an analytical point of view, using as a starting point the

anti-continuum limit (of no-coupling between the sites). Then we generalize our considerations to higher dimensional

settings, showcasing the potentially stable interlacing of more elaborate structures, such as discrete vortices [35] (but

also of vortices with non-vortical structures). We present detailed stability diagrams of such interlaced structures,

and also examine their dynamics when they are found to be unstable.

Our presentation is structured as follows. In section II, we present the model and general mathematical setup. In

section III, we illustrate both analytically and numerically the properties of such structures in 1d settings. In section

IV, we generalize these considerations to a numerical investigation of higher dimensional settings. Finally, in section

V, we summarize our findings and present our conclusions.

MODEL EQUATIONS AND MATHEMATICAL SETUP

We consider a set of coupled DNLS equations

i˙Un+ (g11|Un|2+ g12|Vn|2)Un+ C∆DUn = 0,

i˙Vn+ (g12|Un|2+ g22|Vn|2)Vn+ C∆DVn = 0,(1)

where n is a D-Dimensional index and ∆Dis the discrete Laplacian in D dimensions. We look for stationary solutions

{un}, {vn} through the relation

Un(t) = exp(iΛ1t)un,Vn(t) = exp(iΛ2t)vn. (2)

The dynamical equations (1) then transform into

− Λ1un+ (g11|un|2+ g12|vn|2)un+ C∆Dun = 0,

−Λ2vn+ (g12|vn|2+ g22|vn|2)vn+ C∆Dvn = 0.(3)

The stability is determined in a frame rotating with frequency Λ1for Un(t) and Λ2for Vn(t), i.e., we suppose that

Un(t) = exp(iΛ1t)[un+ ξ(1)

n(t)],Vn(t) = exp(iΛ2t)[vn+ ξ(2)

n(t)].(4)

The small perturbations ξ(k)

n (t), with k = 1,2, can be expressed as

ξ(1)

n(t) = anexp(iλt) + bnexp(−iλ∗t),

leading to the linear stability equations

ξ(2)

n(t) = cnexp(iλt) + dnexp(−iλ∗t),(5)

λJξn= Mnξn+ C(ξn+1+ ξn−1),(6)

with

ξn = (an

b∗

n

cn

d∗

n)T,J =

1

0 −1 0

00

00

000

0

01

0 −1

,(7)

Mn =

K1,n

g11(u2

g12u∗

g12u∗

g11u2

K1,n

n

g12unv∗

g12u∗

K2,n

ng22(v2

ng12unvn

nv∗

g22v2

n)∗

n)∗

nvn g12unvn

nv∗

ng12u∗

nvn

n

n g12unv∗

K2,n

,(8)

K1,n = −Λ + 2g11|un|2+ g12|vn|2− 2C,

K2,n = −Λ + 2g22|vn|2+ g12|un|2− 2C.

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(a)(b) (c)

−10−8−6 −4 −20

n

246810

−1

−0.5

0

0.5

1

un

−10−8−6 −4−20

n

246810

−1

−0.5

0

0.5

1

vn

00.05 0.1 0.15

0

0.2

0.4

0.6

0.8

1

|Re(λ)|

C

00.05 0.10.15

0

0.05

0.1

0.15

0.2

|Im(λ)|

C

00.2 0.40.6 0.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

g12

C

Stable

Hopf+Exponential

Hopf

0.30.350.40.450.50.55

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

H

S

E

E+H

FIG. 1: (a) Profiles of |01? interlaced solitons with g12 = 0.5 and C = 0.15. (b) Dependence on C of the real and imaginary

parts of eigenfrequencies of small perturbations about |01 > with g12 = 0.5. Dashed lines correspond to Lyapunov-Schmidt

predictions of equations (20), and (21). (c) Two-parameter stability diagram in the plane of intersite (C) and inter-component

(g12) coupling, indicating regions of occurance of Hopf bifurcations (H), exponential instability (E), and stability domain (S).

Soliton and vortex solutions are calculated using methods based on the anti-continuous limit. Upon calculating

these solutions at C = 0, we continue them to finite coupling by varying C or other parameters (such as the interspecies

nonlinearity strength g12).

We are interested in interlaced solitons (ISs) in 1D lattices and interlaced vortices (IVs) in 2D and 3D lattices. The

excited sites at C = 0 are equal to ˜ u and ˜ v, except for a phase factor exp(iφ), while unvn= 0 at the corresponding

excited site. These values are

˜ u = 0,

?

Λ1/g11,˜ v = 0,

?

Λ2/g22.(9)

In what follows, we choose Λ1= Λ2≡ Λ and g11= g22= 1. We also choose g12≤ 1 as, for g12> 1 interlaced

solitons and vortices are unstable for every value of C.

ANALYTICAL AND NUMERICAL RESULTS FOR 1D INTERLACED SOLITONS

Existence and stability

We consider interlaced solitons which are labeled by |AB >≡ |A > |B >, where A,B = 0,1,2,.... This number

indicates the “order” of the excited state at the anti-continuous limit, whose phase φ = 0,π is chosen so that the

isolated solitons (i.e. when g12= 0) are stable for any small C. For instance, the ground state |0 > means un= ˜ uδn,0

and the first excited state |1 > will be taken to mean un= ˜ u(δn,1− δn,−1) at the AC limit. Thus, the state |01 >

corresponds to un= ˜ uδn,0, vn= ˜ v(δn,1− δn,−1) and |12 > to un= ˜ u(δn,1− δn,−1), vn= ˜ v(δn,2+ δn,−2) − δn,0.

We first analyze the |01 > state, which is stable for C < C0. At C = C0the ISs become unstable through Hopf

bifurcations (the value of C0differs as a function of the rest of the system parameters such as g12, however the above

scenario is robust). Cascades of this type of bifurcations arise as C increases and, when, C ≥ C1, the ISs become also

exponentially unstable. There is a special region for g12∈ [0.27,0.37] where the system experiences an inverse Hopf

bifurcation recovering the stability in a window. The system becomes unstable again through Hopf bifurcations for

g12∈ [0.27,0.34] and exponential instabilities for g12∈ [0.35,0.37]. Besides, for g12∈ [0.38,0.47] there exist windows

with only exponential instabilities. Fig. 1 illustrates all of the above features, by showcasing a typical example of the

|01 > state, a typical continuation of its principal linearization eigenfrequencies λ, and a full two-parameter diagram

of the stability of this state in the two-parameter plane (C,g12).

For |12 > states, the scenario is essentially similar to the |01 > case, although, in essence, it is considerably simpler

due to the absence of any inverse Hopf bifurcations and restabilization windows. Fig. 2 shows the corresponding

features for |12 >, as Fig. 1 for the |01 > case.

Dynamics of unstable solitons

First, we analyze the dynamics of |01 > ISs. Fig. 3 shows the evolution of a typically unstable (i.e. oscillatory

unstable) |01 > IS with g = 0.2 and C = 0.6. The oscillatory evolution of the instability eventually transforms the

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(a)(b)(c)

−10 −8−6 −4−20

n

246810

−1

−0.5

0

0.5

1

un

−10−8−6−4−20

n

2468 10

−1

−0.5

0

0.5

1

vn

00.050.10.15

0

0.2

0.4

0.6

0.8

1

|Re(λ)|

C

00.050.10.15

0

0.05

0.1

0.15

0.2

|Im(λ)|

C

00.20.40.60.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

g12

C

Stable

Hopf+Exponential

Hopf

FIG. 2: (a) Profiles and (b) dependence on C of the real and imaginary parts of eigenfrequencies of small perturbations of |12?

showing the same features and for the same parameters as in Fig 1. Dashed lines correspond to Lyapunov-Schmidt predictions

of equations (24), and (21). (c) Two-parameter stability diagram in the plane of intersite (C) and inter-component (g12)

coupling.

n

t

|Un|2

−10−50510

0

20

40

60

80

100

120

140

160

180

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

n

t

|Vn|2

−10 −50510

0

20

40

60

80

100

120

140

160

180

0.5

1

1.5

2

2.5

FIG. 3: Time evolution of the density of the two components for a slightly perturbed unstable |01 > IS with g12 = 0.2 and

C = 0.6.

mode into a |00 > state, which is a stable state of the system. The final excited site is typically the same for the

{Un} and {Vn} coordinates, although in some cases (even for the same parameters set), the asymptotic excited site

does not need to be same. However, the amplitude of for the nth site is not identical, i.e. |un| ?= |vn|. In a similar

vein, Fig. 4 shows the evolution of an oscillatory unstable |12 > IS with g = 0.2 and C = 0.4, and, analogously to the

|01 > case, the IS evolves to a |00 > state (although the finally populated site is not the central one of the original

configuration).

n

t

|Un|2

−10−50510

0

50

100

150

200

250

300

350

400

450

0.5

1

1.5

2

n

t

|Vn|2

−10−505 10

0

50

100

150

200

250

300

350

400

450

0

0.5

1

1.5

2

2.5

3

FIG. 4: Time evolution of the density of the two components for a slightly perturbed unstable |12 > IS with g12 = 0.2 and

C = 0.4.

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Perturbation analysis

In this subsection, we attempt to understand in some more details the above observed results of the numerical

computations in connection to the stability properties of the interlaced soliton solutions. More specifically, we evaluate

explicit expressions of the interlaced solitons’ eigenvalues for the configurations discussed above. The method is based

on the expansion in the coupling constant C, in the vicinity of the anti-continuum limit.

In the limit C = 0, as illustrated above, there are two types of solutions, i.e. un= vn= 0, and the non-zero solutions

given by Eqs. (9). In this limit, one can also easily notice that the eigenvalue problem (6) will give

λ = ±Λ, ±Λ(1 − g12/g11), ±Λ(1 − g12/g22)(10)

for the zero solutions and

λ = ±0(11)

for the non-zero solutions (9).

It can be directly inferred from the analysis of the underlying linear problem that the stable eigenvalues λ = ±Λ will

expand creating a band of continuous spectrum when C is increased. Therefore, this eigenvalue will not be discussed

further. The instability for a soliton solution will then be determined by the bifurcation of the remaining eigenvalues.

Let us now first consider the profile of |01 > ISs. It is clear that for finite C the solutions will be deformed from

their AC-limit profile. The leading-order solution up to O(C) is then found to be

u0=

?

Λ

g11+

C

√Λg11,u1= u−1=

?

C

√

Λg11(1−g12/g22),

√Λg22.v0= 0,v1= −v−1=

Λ

g22+

C

(12)

The next step is to consider the stability problem when the coupling is turned on. To the leading order, the

eigenvalue problem of this particular configuration is then given by

MΞ = λσ Ξ, (13)

where

σ = diag(J),Ξ =

ξ−2

ξ−1

ξ0

ξ1

ξ2

,M =

M−2

CId4×4

0

0

0

CId4×4

M−1

CId4×4

0

0

00

0

0

0

0

CId4×4

M0

CId4×4

0

CId4×4

M1

CId4×4

CId4×4

M2

,(14)

and Id4×4is the identity matrix of size 4 × 4.

Since we have expanded unand vnin a power series of C, then it is natural that we also expand all the involved quan-

tities in C, i.e. M = M0+CM1+C2M2+O(C3), Ξ = Ξ0+CΞ1+C2Ξ2+O(C3) and λ = λ0+ Cλ1+ C2λ2+ O(C3).

It can be checked that M0is a singular self-adjoint matrix.

Substituting the expansions to the eigenvalue problems (13) will give us to the leading order

M0Ξ0= λ0σΞ0, (15)

from which one will obtain that λ0is given by Eqs. (10) and (11). In the following, let us first consider the case of

λ0= 0 which are of three pairs, with the corresponding eigenvalues of M0Ξ0= 0 denoted by ej, j = 1,2,3. Therefore,

one can write

Ξ0=

3

?

j=1

cjej.

The next order equation of (13) gives us

M0Ξ1= λ1σ Ξ0− M1Ξ0.(16)

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Using the Fredholm alternative theorem, the above equation will have a solution if the right hand side is orthogonal

to the null space of M0, which it is. Hence, the value of the correction λ1cannot be obtained yet and a solution Ξ1

of (16) can therefore be calculated for any λ1.

The equation of order O(C3) from (6) can be easily deduced to be

M0Ξ2= λ2σΞ0+ λ1σΞ1− M1Ξ1− M2Ξ0.

Projecting the equation above to ej, j = 1,2,3, i.e. basis of the null space of M0, will give us the following eigenvalue

matrix

(17)

−2g11

(g11−g12)Λ0

0

−2g11

(g11−g12)Λ0

−2g11

(g11−g12)Λ

0

−2g11

(g11−g12)Λ

0

c1

c2

c3

= −λ2

1

Λ

c1

c2

c3

,(18)

which can be immediately solve to yield

λ1= ±0, ±0, ±2

?

g11

g11− g12.(19)

This illustrates that there is a pair of eigenvalues bifurcating from zero as given by

λ = ±2C

?

g11

g11− g12

+ O(C2).(20)

The same procedure can be applied to bifurcations of the non-zero eigenvalues. In this case, the calculation is even

simpler as applying the Fredholm alternative to the O(C) equation of (13) already gives us a solvability condition

from which we obtain that bifurcating eigenvalues are

λ = ±(1 − g12/g22)(Λ + 2C), ±(1 − g12/g11)(Λ + 2C), (21)

The above procedure can also be similarly and immediately applied to the configuration |12 > ISs. The only

difference is that for that solution one will obtain a stability matrix M of size 28 × 28.

For |12 >ISs, we can obtain the solution in a power series of C as

?

v0= −v2= −v−2= −

u0= 0,u1= −u−1=

Λ

g11+

C

√Λg11,

C

√Λg22,

u2= −u−2=

v1= v−1= 0.

C

√Λg11(1−g12/g22),

?

Λ

g22−

(22)

Continuing to finding the eigenvalues, we will also immediately obtain that in place of (18), one will obtain the

following eigenvalue problem

−2g11

(g11−g12)Λ

0

−2g11

(g11−g12)Λ

0

0

0

−2g11

(g11−g12)Λ

0

−4g11

(g11−g12)Λ

0

−2g11

(g11−g12)Λ

00

0

−2g22

(g22−g12)Λ

0

−2g22

(g22−g12)Λ

0

−2g22

(g22−g12)Λ

0

−2g22

(g22−g12)Λ

0

−2g11

(g11−g12)Λ

0

−2g11

(g11−g12)Λ

c1

c2

c3

c4

c5

= −λ2

1

Λ

c1

c2

c3

c4

c5

, (23)

from which we can obtain eigenvalues bifurcating from zero as

λ = ±

?

2

1 − g12/g11C, ±

?

6

1 − g12/g11C, ±

?

4

1 − g12/g22C.(24)

Bifurcations from the non-zero eigenvalues for this case can also be shown to yield Eq. (21).

The above analytical expressions give us a detailed handle on the dependence of the relevant eigenalues on the

system parameters. Comparisons of the analytical results obtained here with the numerical ones are presented in

Figs. 1-2 where one can see that the analytical expressions are in relatively good agreement with the numerical

results. It should be noted that although such analytical considerations are procedurally straightforward to generalize

in higher dimensions, the relevant calculations are extremely tedious and will thus not be pursued here. Instead,

we now turn to numerical computations to showcase the existence and potential stability of interlaced solitons and

vortices in higher-dimensional settings.

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(a) (b)(c)

m

n

Re(un,m)

−5 05

−5

0

5

−1

−0.5

0

0.5

1

m

n

Re(vn,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

m

n

Im(un,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

m

n

Im(vn,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

0 0.050.10.15

0

0.2

0.4

0.6

0.8

1

|Re(λ)|

C

00.05 0.10.15

0

0.1

0.2

0.3

0.4

0.5

|Im(λ)|

C

0 0.20.40.6 0.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

g12

C

Stable

Hopf+Exponential

Hopf

FIG. 5: (a) Plot of real (top) and imaginary (bottom) parts of interlaced vortices with g12 = 0.5 and C = 0.1. (b) Dependence

on C of the real and imaginary parts of eigenfrequencies of small perturbations about such solutions with g12 = 0.5 (b)

Two-parameter stability diagram in the plane of intersite (C) and inter-component (g12) coupling.

(a)(b)(c)

m

n

Re(un,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

m

n

Re(vn,m)

−505

−5

0

5

0.2

0.4

0.6

0.8

1

m

n

Im(un,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

m

n

Im(vn,m)

−505

−5

0

5

−1

−0.5

0

0.5

1

00.05 0.10.15

0

0.2

0.4

0.6

0.8

1

|Re(λ)|

C

00.050.10.15

0

0.05

0.1

0.15

0.2

|Im(λ)|

C

00.2 0.40.60.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

g12

C

Stable

Hopf+Exponential

Hopf

Hopf

FIG. 6: (a) Plot of real (top) and imaginary (bottom) parts and (b) dependence on C of the real and imaginary parts of

eigenfrequencies of small perturbations about interlaced soliton-vortex solutions, showing the same features and for the same

parameters as in Fig. 5. (c) The corresponding two-parameter stability diagram.

NUMERICAL RESULTS FOR INTERLACED STRUCTURES IN HIGHER DIMENSIONS

For the case of 2D lattices, we consider two different interlaced structures. On the one hand, we examine interlaced

vortices (IVs) whose configurations in the AC limit are given by u0,1 = ˜ u, u0,−1 = −˜ u, u1,0 = i˜ u, u−1,0 = −i˜ u;

v1,1= −i˜ v, v1,−1= ˜ v, v−1,−1= i˜ v, v−1,1= −˜ v. On the other hand, we also study a discrete soliton interlaced with

a vortex (IVSs) whose configurations in the AC limit are given by u0,1= ˜ u, u0,−1= −˜ u, u1,0= i˜ u, u−1,0= −i˜ u,

v0,0= 1.

IVs experience a set of bifurcation scenaria which are qualitatively similar to those of the |12 > ISs. IVSs experience

the same scenario as well, with the basic difference that they appear to exist for all C’s (within the range examined

i.e., up to C = 2) for g ≤ 0.4. Also, notably, the IVSs experience solely Hopf bifurcations in a fairly small region

inside the exponential+Hopf region. Figs. 5 and 6 summarize the corresponding findings in a way similar as for the

1d configurations, presenting not only typical profiles of the modes, but also typical mono-parametric continuations,

as well as their full two-parameter stability diagram in the space of inter-site and inter-species coupling.

In the case of 3D lattices, we consider two interlaced vortices conjoined in the shape of a cube. In the AC limit, this

cube is given by u−1,1,1= ˜ u, u1,−1,1= −˜ u, u−1,−1−1= i˜ u, u1,1,−1= −i˜ u; v1,−1,−1= ˜ v, v−1,1,−1= −˜ v, v1,1,1= i˜ v,

v−1,−1,1= −i˜ v. The structure is stable near the AC limit, with the size of the window of stability diminishing as g12

approaches 1, and with instability setting in via Hopf bifurcations. In the 2-parameter continuation figure shown in

Fig. 7, the coupling is only continued to C = 0.75, but it is observed that for values between g12≈ 0.703 and g12= 1,

the instability further degenerates into Hopf and exponential instabilities. It should also be noted that within the

region of instability, there some exist isolated points or very narrow regions where inverse Hopf bifurcations may be

observed, which have been omitted from the graph for clarity. Let us note in passing here that the interlaced vortices

in the “vortex cube” shown in Fig. 7 are perhaps not the prototypical interlaced structure that one would expect in

3D; instead one might expect a structure where each vortex is confined in a diagonal plane within the cube (with the

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2

−2

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2

−2

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2

n

m

Re(un,m,l)

l

−20

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−2

0

2

−2

0

2

n

m

Im(un,m,l)

l

−20

2

−2

0

2

−2

0

2

n

m

Re(vn,m,l)

l

−20

2

−2

0

2

−2

0

2

n

m

Im(vn,m,l)

l

0 0.20.40.60.81

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

g12

C

Stable

Hopf

Hopf+

Exponential

00.05 0.1

C

0.150.2

0

0.1

0.2

|Re(λ)|

0.050.10.150.2

0.05

0.1

0.15

C

|Im(λ)|

FIG. 7: (a) The top set of four panels show an interlaced cube (g = 0.85) in a grid of size 11×11×11 that has been continued

to coupling C = 0.20. The level contours shown correspond to Re(un,m,l) = Re(vn,m,l) = ±0.5max{un,m,l}, in blue and

red (dark gray and gray, in the black-and-white version) respectively, while the imaginary ones, Im(un,m,l) = Im(vn,m,l) =

±0.5max{un,m,l}, are shown by green and yellow (light and very light gray, in the black-and-white version) respectively. The

bottom panel shows the real eigenfrequencies of small perturbation. (b) The top panel shows the stability diagram, while the

bottom panel shows the imaginary eigenfrequencies of small perturbation.

two such planes intersecting transversally). We were, however, unable to trace such a structure even in the vicinity

of the anti-continuum limit.

Dynamics of unstable structures

The dynamics of the oscillatory unstable IVs in 2D lattices with g12= 0.2 and C = 0.3 is shown if Fig. 8. The

evolution results in the transformation of the original structure into single-peaked or multi-peaked solitons. Excited

peaks do not coincide for Unand Vn. The vorticity of each vortex is lost. Fig. 9 shows the dynamics of an oscillatorily

unstable interlaced vortex-soliton structure with g12= 0.2 and C = 0.45. This mode evolves spontaneously towards

single-peaked solitons. The excited peaks are in the same site in both lattices in this example.

Dynamics of the interlaced cube with g12= 0.85 is shown in Fig. 10. Here the coupling is continued to C = 0.6.

This is well past the threshold of stability for this value of g12, and takes the configuration into the region of both

exponential and oscillatory instabilites. It is observed that when a peturbation of magnitude 0.01 is applied, only a

single site survives for long times (in this case for Vn).

Although these are prototypical results of the dynamical evolution, which we have generically observed to lead to

less elaborate (and often purely single-peaked) structures in this setting, it should be stressed that the specific details

of the unstable dynamical evolution of each structure depend considerably on the values of the parameters, as well as

partially on the type/strength of the perturbation.

CONCLUSIONS AND FUTURE CHALLENGES

In the present work, we have illustrated the possibility to successfully interlace structures which are stable in each one

of the components (either simple ones, such as single site solitary waves, or more elaborate ones, such as bound states

and vortices) in order to produce stable multi-component interlaced solitons/vortices. We have continued the resulting

structures from the anti-continuum limit of no inter-site coupling to finite coupling and illustrated the intervals of

stability, as well as the ones of both exponential and oscillatory (Hopf) instabilities. We have given detailed two-

parameter diagrams of the stable ranges of the solutions as a function of the inter-site and inter-component couplings.

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t=65

0.5

1

1.5

2

t=95

1

2

3

t=125

1

2

3

t=155

1

2

3

t=185

1

2

3

t=215

1

2

3

t=65

0.5

1

1.5

t=95

0.5

1

1.5

2

t=125

0.5

1

1.5

2

t=155

0.5

1

1.5

2

t=185

0.5

1

1.5

2

t=215

0.5

1

1.5

2

2.5

FIG. 8: Snapshots showing (a) |Un(t)|2and (b) |Vn(t)|2for unstable IVs with g12 = 0.2 and C = 0.3.

(a) (b)

t=85

t=90

t=95

t=100

t=105

t=110

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

1

2

3

0.5

1

1.5

2

1

2

3

4

t=85

0.5

1

1.5

2

t=90

0.5

1

1.5

2

t=95

0.5

1

1.5

2

t=100

0.5

1

1.5

2

t=105

0.5

1

1.5

2

2.5

t=110

0.5

1

1.5

FIG. 9: Snapshots showing (a) |Un(t)|2and (b) |Vn(t)|2for unstable interlaced vortex-solitons with g12 = 0.2 and C = 0.45.

These revealed that the linear stability of the interlaced structures necessitates sufficiently weak coupling (typically

no larger than 0.4, with the relevant range decreasing as the inter-component interaction is increased) and sufficiently

weak inter-component interaction (i.e., g12 < 1). Finally, we examined the dynamical evolution of the instability

of such interlaced structures, which typically resulted in the destruction of the waveforms, in favor of simpler, more

stable dynamical patterns.

Nevertheless, there is still a number of important open questions for future consideration. For instance, it would be

particularly interesting to examine whether it would be possible for the inter-component coupling to actually stabilize

structures that are dynamically unstable in the single-component setting. Also, it would be useful to possess a

systematic classification of the solutions (interlaced and non-interlaced ones) available in the multi-component system

setting, similarly to the one-component classifications of [34, 35]. Such efforts are currently underway and will be

reported in future publications.

[1] H.S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).

[2] R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg and Y. Silberberg, Phys. Rev. Lett. 83, 2726-2729 (1999); H.

S. Eisenberg, Y. Silberberg, R. Morandotti and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).

[3] D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).

[4] D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817-823 (2003); A. A. Sukhorukov, Yu. S. Kivshar, H.

S. Eisenberg, and Y. Silberberg, IEEE J. Quant. Elect. 39, 31 (2003).

[5] S. Aubry, Physica 103D, 201 (1997); S. Flach and C. R. Willis, Phys. Rep. 295, 181 (1998);

[6] N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev Phys. Rev. E 66, 046602 (2002).

[7] J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).

[8] J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).

[9] J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).

[10] J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Stud. Appl. Math. 113, 389 (2004).

[11] J. Yang, I. Makasyuk, P. G. Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, Phys. Rev. Lett.

Page 10

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t=12

2

−2

0

m

2

−2

0

2

n

t=4

l

−20

t=45

2

−2

0

m

2

−2

0

2

n

t=9

l

−20

2

−2

0

m

2

−2

0

2

n

l

−20

2

−2

0

m

2

−2

0

2

l

n

−20

t=12

2

−2

0

m

2

−2

0

2

n

t=4

l

−20

t=45

2

−2

0

m

2

−2

0

2

n

t=9

l

−20

2

−2

0

m

2

−2

0

2

n

l

−20

2

−2

0

m

2

−2

0

2

n

l

FIG. 10: Snapshots showing evolution of (a) Un(t) and (b) Vn(t) for the interlaced cube with g12 = 0.85 in a grid of size

11 × 11 × 11 where the coupling has been continued to C = 0.6. All iso-contour plots are defined as Re(un,m,l) = Re(vn,m,l) =

±0.75 = Im(un,m,l) = Im(vn,m,l), where in the figure, dark gray (blue) and gray (red) colors pertain to iso-contours of the

real part of the solutions, while the light gray (green) and very light gray (yellow) colors correspond to the iso-contours of the

imaginary part. The configuration was pertubed by a random noise of amplitude 0.01 in order to expedite the onset of the

instability.

94, 113902 (2005).

[12] F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506 (2005).

[13] D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Yu. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett.

92, 123903 (2004).

[14] J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett. 92,

123904 (2004).

[15] Y.V. Kartashov, V.A. Vysloukh and L. Torner, Phys. Rev. Lett. 93, 093904 (2004); X. Wang, Z. Chen, and P. G. Kevrekidis,

Phys. Rev. Lett. 96, 083904 (2006).

[16] D. Tr¨ ager, R. Fischer, D.N. Neshev, A.A. Sukhorukov, C. Denz, W. Kr´ olikowski and Yu.S. Kivshar, Optics Express 14,

1913 (2006).

[17] G. Bartal, O. Manela, O. Cohen, J.W. Fleischer and M. Segev, Phys. Rev. Lett. 95, 053904 (2005).

[18] H. Trompeter, W. Kr´ olikowski, D.N. Neshev, A.S. Desyatnikov, A.A. Sukhorukov, Yu.S. Kivshar, T. Pertsch, U. Peschel

and F. Lederer, Phys. Rev. Lett. 96, 053903 (2006).

[19] O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev and D.N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).

[20] C.R. Rosberg, D.N. Neshev, A.A. Sukhorukov, W. Krolikowski and Yu.S. Kivshar, Opt. Lett. 32, 397 (2007).

[21] B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D.N. Christodoulides and J.W. Fleischer, Nature 440, 1166 (2006).

[22] T. Schwartz, G. Bartal, S. Fishman and M. Segev, Nature 446, 52 (2007).

[23] L. Hadzievski, A. Maluckov, M. Stepi´ c and D. Kip, Phys. Rev. Lett. 93, 033901 (2004); L. Hadzievski, A. Maluckov and

M. Stepi´ c, Opt. Express 15, 5687 (2007).

[24] E.P. Fitrakis, P.G. Kevrekidis, H. Susanto and D.J. Frantzeskakis, Phys. Rev. E 75, 066608 (2007); V.M. Rothos, H.E.

Nistazakis, P.G. Kevrekidis and D.J. Frantzeskakis, J. Phys. A: Math. Theor. 42, 025207 (2009).

[25] P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-Gonz´ alez (eds). Emergent Nonlinear Phenomena in Bose-Einstein

Condensates: Theory and Experiment, Springer-Verlag (Heidelberg, 2008).

[26] J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti and J.S. Aitchison, Phys. Rev. Lett.

91, 143907 (2003)

[27] Z. Chen, J. Yang, A. Bezryadina, and I. Makasyuk, Opt. Lett. 29, 1656 (2004).

[28] C. J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell and C.E. Wieman, Phys. Rev. Lett. 78, 586 (1997)

[29] K.M. Mertes, J.W. Merrill, R. Carretero-Gonz´ alez, D.J. Frantzeskakis, P.G. Kevrekidis, and D.S. Hall, Phys. Rev. Lett.

99, 190402 (2007).

[30] D.M. Stamper-Kurn, M.R. Andrews, A.P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger and W. Ketterle, Phys. Rev.

Lett. 80, 2027 (1998)

[31] G. Modugno, G. Ferrari, G. Roati, R.J. Brecha, A. Simoni and M. Inguscio, Science 294, 1320 (2001)

[32] M. Mudrich, S. Kraft, K. Singer, R. Grimm, A. Mosk and M. Weidem¨ uller, Phys. Rev. Lett. 88, 253001 (2002).

[33] O. Morsch and E. Arimondo, in Dynamics and Thermodynamics of Systems with Long-Range Interactions, T. Dauxois, S.

Ruffo, E. Arimondo and M. Wilkens (Eds.), Springer (Berlin 2002), pp. 312-331.

[34] G. L. Alfimov, V. A. Brazhnyi, V. V. Konotop, Physica D 194, 127 (2004). D.E. Pelinovsky, P.G. Kevrekidis, and D.J.

Frantzeskakis, Physica D 212, 1 (2005).

[35] D.E. Pelinovsky, P.G. Kevrekidis, and D.J. Frantzeskakis, Physica D 212, 20 (2005); M. Lukas, D. Pelinovsky and P.G.

Kevrekidis, Physica D 237, 339 (2008).