Interlaced solitons and vortices in coupled DNLS lattices
ABSTRACT In the present work, we propose a new set of coherent structures that arise in nonlinear dynamical lattices with more than one component, namely interlaced solitons. In the anticontinuum limit of uncoupled sites, these are waveforms whose one component has support where the other component does not. We illustrate systematically how one can combine dynamically stable unary patterns to create stable ones for the binary case of twocomponents. For the onedimensional setting, we provide 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 they are exponentially or oscillatorily unstable, to structures with a smaller number of participating sites.

Article: Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Arbitrary Order
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ABSTRACT: Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lam\'e polynomials of arbitrary order. The models discussed are (i) coupled Salerno model, (ii) coupled AblowitzLadik model, (iii) coupled $\phi^4$ model, and (iv) coupled $\phi^6$ model. In all these cases we show that the coefficients of the Lam\'e polynomials are such that the Lam\'e polynomials can be reexpressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.Pramana 11/2011; 79(3). · 0.56 Impact Factor  SourceAvailable from: Vassilis Rothos[Show abstract] [Hide abstract]
ABSTRACT: In the present work, we generalize earlier considerations for intrinsic localized modes consisting of a few excited sites, as developed in the onecomponent discrete nonlinear Schrodinger equation model, to the case of twocomponent systems. We consider all the different combinations of "up" (zero phase) and "down" ({\pi} phase) site excitations and are able to compute not only the corresponding existence curves, but also the eigenvalue dependences of the small eigenvalues potentially responsible for instabilities, as a function of the nonlinear parameters of the model representing the self/cross phase modulation in optics and the scattering length ratios in the case of matter waves in optical lattices. We corroborate these analytical predictions by means of direct numerical computations. We infer that all the modes which bear two adjacent nodes with the same phase are unstable in the two component case and the only solutions that may be linear stable are ones where each set of adjacent nodes, in each component is out of phase.Mathematics and Computers in Simulation 02/2012; · 0.84 Impact Factor  SourceAvailable from: Boris A. Malomed[Show abstract] [Hide abstract]
ABSTRACT: It is known that, in continuous media, composite solitons with hidden vorticity, which are built of two mutually symmetric vortical components whose total angular momentum is zero, may be stable while their counterparts with explicit vorticity and nonzero total angular momentum are unstable. In this work, we demonstrate that the opposite occurs in discrete media: hidden vortex states in relatively small ring chains become unstable with the increase of the total power, while explicit vortices are stable, provided that the corresponding scalar vortex state is also stable. There are also stable mixed states, in which the components are vortices with different topological charges. Additionally, degeneracies in families of composite vortex modes lead to the existence of longlived breather states which can exhibit vortex charge fipping in one or both components.Journal of Optics 12/2012; 15(4). · 2.01 Impact Factor
Page 1
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, E41011 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 010034515, 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 anticontinuum 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 twocomponents.
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 PeierlsNabarropotential 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], twodimensional (2D) Bloch oscillations and LandauZener tunneling [18], wave
formation in honeycomb [19], hexagonal [20] and quasicrystalline 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 BoseEinstein condensates (BECs) trapped in periodic potentials. There, once
again, a reduction of the relevant model can be formulated in the tightbinding approximation within the meanfield
limit, reducing the socalled GrossPitaevskii 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, multicomponent 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 morecomponent generalizations are certainly possible) systems of DNLS equations. We
Page 2
2
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 viceversa. In the onedimensional 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
anticontinuum limit (of nocoupling 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 nonvortical 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+ (g11Un2+ g12Vn2)Un+ C∆DUn = 0,
i˙Vn+ (g12Un2+ g22Vn2)Vn+ C∆DVn = 0,(1)
where n is a DDimensional 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+ (g11un2+ g12vn2)un+ C∆Dun = 0,
−Λ2vn+ (g12vn2+ g22vn2)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 + 2g11un2+ g12vn2− 2C,
K2,n = −Λ + 2g22vn2+ g12un2− 2C.
Page 3
3
(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 LyapunovSchmidt
predictions of equations (20), and (21). (c) Twoparameter stability diagram in the plane of intersite (C) and intercomponent
(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 anticontinuous 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 anticontinuous 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 twoparameter diagram
of the stability of this state in the twoparameter 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
Page 4
4
(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 LyapunovSchmidt predictions
of equations (24), and (21). (c) Twoparameter stability diagram in the plane of intersite (C) and intercomponent (g12)
coupling.
n
t
Un2
−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
Vn2
−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
Un2
−10−50510
0
50
100
150
200
250
300
350
400
450
0.5
1
1.5
2
n
t
Vn2
−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.
Page 5
5
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 anticontinuum limit.
In the limit C = 0, as illustrated above, there are two types of solutions, i.e. un= vn= 0, and the nonzero 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 nonzero 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 AClimit profile. The leadingorder 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 selfadjoint 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)
Page 6
6
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 nonzero 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 nonzero 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. 12 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 higherdimensional settings.
Page 7
7
(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)
Twoparameter stability diagram in the plane of intersite (C) and intercomponent (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 solitonvortex solutions, showing the same features and for the same
parameters as in Fig. 5. (c) The corresponding twoparameter 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 monoparametric continuations,
as well as their full twoparameter stability diagram in the space of intersite and interspecies 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 2parameter 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
Page 8
8
(a)(b)
−20
2
−2
0
2
−2
0
2
n
m
Re(un,m,l)
l
−20
2
−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 blackandwhite 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 blackandwhite 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 anticontinuum 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 singlepeaked or multipeaked 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 vortexsoliton structure with g12= 0.2 and C = 0.45. This mode evolves spontaneously towards
singlepeaked 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 singlepeaked) 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 multicomponent interlaced solitons/vortices. We have continued the resulting
structures from the anticontinuum limit of no intersite 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 intersite and intercomponent couplings.
Page 9
9
(a)(b)
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 vortexsolitons 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 intercomponent interaction is increased) and sufficiently
weak intercomponent 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 intercomponent coupling to actually stabilize
structures that are dynamically unstable in the singlecomponent setting. Also, it would be useful to possess a
systematic classification of the solutions (interlaced and noninterlaced ones) available in the multicomponent system
setting, similarly to the onecomponent classifications of [34, 35]. Such efforts are currently underway and will be
reported in future publications.
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(a)(b)
−20
t=12
2
−2
0
m
2
−2
0
2
n
t=4
l
−20
t=45
2
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0
m
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0
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n
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l
−20
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0
m
2
−2
0
2
n
l
−20
2
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0
m
2
−2
0
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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
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2
n
t=9
l
−20
2
−2
0
m
2
−2
0
2
n
l
−20
2
−2
0
m
2
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0
2
n
l
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real part of the solutions, while the light gray (green) and very light gray (yellow) colors correspond to the isocontours 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.
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