P. G. Kevrekidis and D.E. Pelinovsky- Discrete vector on-site vortices

8/3/2019 P. G. Kevrekidis and D.E. Pelinovsky- Discrete vector on-site vortices

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doi: 10.1098/rspa.2006.1693, 2671-26944622006Proc. R. Soc. A

P.G Kevrekidis and D.E PelinovskyDiscrete vector on-site vortices

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Discrete vector on-site vortices

BY P. G. KEVREKIDIS1 AN D D. E. PELINOVSKY2,*

1Department of Mathematics, University of Massachusetts, Amherst,MA 01003-4515, USA

2Department of Mathematics, McMaster University, Hamilton,Ontario L8S 4K1, Canada

We study discrete vortices in coupled discrete nonlinear Schrodinger equations. We focuson the vortex cross configuration that has been experimentally observed inphotorefractive crystals. Stability of the single-component vortex cross in the anti-continuum limit of small coupling between lattice nodes is proved. In the vector case, weconsider two coupled configurations of vortex crosses, namely the charge-one vortex inone component coupled in the other component to either the charge-one vortex (forminga double-charge vortex) or the charge-negative-one vortex (forming a, so-called, hidden-charge vortex). We show that both vortex configurations are stable in the anti-continuum limit, if the parameter for the inter-component coupling is small and both ofthem are unstable when the coupling parameter is large. In the marginal case of thediscrete two-dimensional Manakov system, the double-charge vortex is stable while thehidden-charge vortex is linearly unstable. Analytical predictions are corroborated withnumerical observations that show good agreement near the anti-continuum limit, but

gradually deviate for larger couplings between the lattice nodes.

Keywords: discrete nonlinear Schrodinger equations; vortices;

persistence and stability; LyapunovSchmidt reductions

1. Introduction

In the past few years, the developments in the nonlinear optics of photorefractivematerials (Fleischer et al. 2005) and of BoseEinstein condensates (BECs) in

optical lattices (Brazhnyi & Konotop 2004) have stimulated an enormousamount of theoretical, numerical and experimental activity in the area of discretenonlinear Hamiltonian systems. A particular focus in this effort has been drawnto the prototypical lattice model of the discrete nonlinear Schrodinger (DNLS)equation (Kevrekidis et al. 2001). The latter, either as a tight binding limit(Alfimov et al. 2002) or as a generic discrete nonlinear envelope wave equation(Kivshar & Peyrard 1992) plays a key role in unveiling the relevant dynamicswithin the appropriate length and time scales.

One of the principal directions of interest in these lattice systems consists ofthe effort to analyse the main features of their localized solutions. In the

particular case of two spatial dimensions, such structures can be discrete gap

Proc. R. Soc. A (2006) 462, 26712694

doi:10.1098/rspa.2006.1693

Published online 4 April 2006

* Author for correspondence ([email protected]).

Received 23 November 2005Accepted 13 February 2006 2671 q 2006 The Royal Society

http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/


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solitons (Christodoulides & Joseph 1988; Kivshar 1993) or discrete vortices (i.e.structures that have topological charge over a discrete contour; Malomed &Kevrekidis 2001; Yang & Musslimani 2003). The study of these types of coherentstructures has made substantial leaps of progress in the past two years with thenumerical and experimental observation of regular discrete solitons (Fleischer

et al. 2003a,b), dipole solitons (Yang et al. 2004), soliton-trains (Chen et al.2004a), soliton-necklaces (Yang et al. 2005) and vector solitons (Chen et al.2004b) in photorefractive crystals and experimental discovery of robust discretevortex states (Neshev et al. 2004; Fleischer et al. 2004).

On the other hand, the recent years were marked by the experimentaldevelopments in soft condensed-matter physics of BECs. Among the importantrecent observations, one can single out the experimental illustration of the dark(Burger et al. 1999), bright (Strecker et al. 2002) and gap (Eiermann et al. 2004)solitons in quasi-one-dimensional BECs. The experimental capabilities seem tobe on the verge of producing similar structures in a two-dimensional context

(Greiner et al. 2001).In both of the above contexts (nonlinear optics and atomic physics), multi-component systems were recently studied due to their relevance to applications.In particular, the first observations of discrete vector solitons in nonlinearwaveguide arrays were reported in Meier etal. (2003), while numerous experimentswith BECs were directed towards studies of mixtures of different spin states of87Rb(Myatt et al. 1997) or 23Na (Stamper-Kurn et al. 1998) and even ones of differentatomic species such as 41KK87Rb (Modugno et al. 2001) and 7LiK133Cs (Mudrichet al. 2002). While the above BEC experiments did not include the presence of anoptical lattice, the addition of an external optical potential could be manufacturedwithin the present experimental capabilities (Brazhnyi & Konotop 2004).

It is the purpose of the present work to address these recent features of thephysical experiments, namely discrete systems with multiple components. Inparticular, we aim at addressing the fundamental issue of how localizedexcitations are affected by the presence of two components which are coupled(nonlinearly) to each other. While our results will be presented for the specificexample of two coupled DNLS equations with cubic nonlinearities, we believethat similar features persist in a variety of other models. We should note herethat rather few studies have focused on the two-dimensional vector generaliz-ation of the DNLS equation (Ablowitz & Musslimani 2002; Hudock et al. 2003;Vicencio et al. 2004). To the best of our knowledge, these earlier studies did not

address vortices in coupled discrete systems.For vortices in coupled systems, a number of interesting questions emerges

concerning the stability of particular vortex configurations (e.g. the so-calledvortex cross; Neshev et al. 2004; Fleischer et al. 2004) including the case of equalcharges in both components and the case of opposite charges between the twocomponents. The former state has a double vortex charge, while the latter has ahidden vortex charge. It has been shown for the continuous NLS equation withcubic-quintic (Desyatnikov et al . 2005) and saturable (Ye et al . 2004)nonlinearities that these two states have different stability windows.

In the present setting, we examine the stability of such vortex structures in

the discrete case, both analytically and numerically. We use the method ofLyapunovSchmidt (LS) reductions developed earlier in Pelinovsky et al.(2005a,b). This method allows for direct analytical calculations of eigenvalues

P. G. Kevrekidis and D. E. Pelinovsky2672

Proc. R. Soc. A (2006)



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of the linear stability problem as functions of the system parameters (such asthe coupling between adjacent lattice sites and the coupling between thetwo components).

Our presentation is structured as follows. In 2, we introduce the setup andthe vortex cross configurations. In 3, we study the stability of such

configurations in the one-component model. In 4, we generalize the vortexcross configuration to the two-component case and compare our results withnumerical computations of the parameter continuations. In 5, we deal with aspecial Manakov case of the system of two DNLS equations. Finally, in 6, wesummarize our findings. Appendix A presents technical details for the case of thesingle-component vortex cross.

2. Setup

We write the coupled system of DNLS equations in the form:

i _un;mCeunC1;mCunK1;mCun;mC1Cun;mK1Cjun;mj2Cbjvn;mj2un;mZ 0; 2:1

i _vn;mCevnC1;mCvnK1;mCvn;mC1Cvn;mK1Cbjun;mj2C jvn;mj2vn;mZ 0; 2:2where bR0 and eR0. Parameter b is often referred to as the cross-phasemodulation coefficient in optics. The self-phase modulation has been set to unityin the systems (2.1) and (2.2). As is discussed in detail in Hudock et al. (2003; seealso references therein), typical values ofb in the optical setting are bZ2=3 fortwo linear mutually orthogonal polarization modes in each waveguide and bZ2

for two circular polarizations (or two different carrier wavelengths). On the otherhand, for binary condensates this coefficient is of the order of unity (Myatt et al.1997) producing at bZ1 the discrete (non-integrable) analogue of the well-knowncontinuum integrable Manakov system. Hence, we will particularly focus onthese values of b in our numerical results in what follows, even though ouranalysis will be kept as general as possible.

Localized modes of the coupled systems (2.1) and (2.2) take the form,

un;mtZfn;meit; vn;mtZjn;meiut; 2:3where u is a parameter of time-periodic solutions and

fn;m;jn;m

satisfy the

system of nonlinear difference equations,

1Kjfn;mj2Kbjjn;mj2fn;mZ efnC1;mCfnK1;mCfn;mC1Cfn;mK1; 2:4

uKbjfn;mj2Kjjn;mj2jn;mZ ejnC1;mCjnK1;mCjn;mC1Cjn;mK1: 2:5We are interested in a particular vortex solution, called the vortex cross. Anexample of this solution is obtained numerically for bZ2=3, uZ1 and eZ0:1 andit is shown in figure 1. Let us consider the diagonal square discrete contour on thegrid n; m2Z2,

S0ZfK1; 0

;

0;K1

;

1; 0

;

0; 1

g3Z

2;2:6

enumerated in the same order by jZ1; 2; 3; 4. We shall assume that the vortexcross of figure 1 bifurcates from the limiting solution at the anti-continuum

2673Discrete vector on-site vortices




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limit eZ0:

f0n;mZ

aeiqj; n; m2S0;0; n; m;S0;

j0n;mZ

b einj; n; m2S0;0; n; m;S0;

((2:7

where the set of phase parameters fqj; njg4jZ1 is yet to be determined, while theset of amplitude parameters (a, b) is determined from solutions of the system

a2Cbb2Z 1; ba2Cb2Zu: 2:8

When bs1, there exists a unique solution of the system (2.8),

a2Z1Kbu

1Kb2; b2Z

uKb

1Kb2: 2:9

The solution is meaningful only if a2O0 and b2O0, which define the domainof existence

minb; bK1%u%maxb; bK1: 2:10When bZ1, the domain of existence shrinks into the line uZ1 and thesolution of the system (2.8) forms a one-parameter family,

aZ cos d; bZ sin d; d20; 2p: 2:11The vortex cross, if it exists, is defined by the phase configurations along thediscrete contour S0,

qjZpjK1

2; njZG

pjK12

; jZ 1; 2; 3; 4: 2:12

The upper sign corresponds to the (1, 1) coupled state called the double-chargevortex, while the lower sign corresponds to the (1,K1) coupled state called the

hidden-charge vortex. Persistence and stability of the vortex configurations(2.7), (2.9) and (2.12) are addressed separately in the cases bZ0, 0!b!1,bZ1 and bO1.

m

2

0

2

0

0.5

5

0

5

2

0

2

m

m

m

n

2 0 2

2

0

20

0.5

n

5 0 5

5

0

5

2

0

2

m

2

0

2

5

0

5

m

m

m

n

2 0 2

2

0

2

n

5 0 5

5

0

5

Figure 1. The contour plots show the amplitude and phase (left and right panels, respectively) ofthe two components (top and bottom, respectively) for a (1, 1) (left four subplots) and a (1, K1)(right four subplots) vortex configuration, in the case of bZ2=3, uZ1 and eZ0:1.





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3. Scalar vortex cross

We apply the method of LS reductions developed in Pelinovsky et al. (2005b) tothe scalar nonlinear difference equation

1K

jFn;mj2

Fn;mZ

eFnC1;mCFnK1;m

CFn;mC1

CFn;mK1: 3:1

This scalar equation corresponds to the reduction jn;mZ0,cn; m2Z2 of thesystems (2.4) and (2.5). Local existence of a single-component vortex cross in thescalar problem (3.1) is proved in appendix A for small values of e (on the basis ofproposition 2.9 in Pelinovsky et al. (2005b)). This result is formulated as follows.

Proposition 3.1. There exists a unique (up to the gauge invariance)continuation in e of the limiting solution at eZ0:

F0n;mZ

eiqj; n; m2S0;

0; n; m;

S0

;( 3:2

where S0 is given by equation (2.6) and the values of qj are given by equation(2.12). The family of vortex solutions Fn;me, n; m2Z2 is a smooth (realanalytic) function of e.

To address spectral stability of the vortex cross in the time-evolution of thesingle-component DNLS equation, we consider the linearization problem with theexplicit formula

un;mtZ eitFn;mCan;meltCbn;melt;and derive the linear eigenvalue problem from the DNLS equation,

1K2jFn;mj2an;mKF2n;mbn;mKeanC1;mCanK1;mCan;mC1Can;mK1Z ilan;m;3:3

1K2jFn;mj2bn;mK F2n;man;mKebnC1;mCbnK1;mCbn;mC1Cbn;mK1ZKilbn;m;3:4

where l is an eigenvalue and an;m; bn;m are components of an eigenvector.Symbolically, we write the linear eigenvalue problem as

He4Z ils4; 3:5where He is the linearized Jacobian matrix for the systems (3.1), s is a diagonalmatrix of (1,K1) and 4 is an eigenvector consisting of an;m; bn;m. The lineareigenvalue problem for the limiting solution Fn;mZF

0n;m at eZ0 has a set of

double zero eigenvalues with the eigenvectors ej and generalized eigenvectors ej,

such that H0ejZ0 and H0ejZ2isej, where H0ZH0. The index jenumerates the set S0 and the eigenvectors ej and ej have non-zero componentsonly at the corresponding nodes of the set S0,

ejZ ie

iqj

KeKiqj

!; ejZ

eiq

j

eKiqj

!: 3:6





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The kernel of He for es0 includes at least one eigenfunction,

4n;mZFn;m

KFn;m

!; n; m2Z2; 3:7

which follows from the gauge invariance of the DNLS equation with respect torotation of the complex phase in Fn;m, Fn; m2Z2. It is easy to show that ageneralized kernel for zero eigenvalue is non-empty as it includes a solution of theinhomogeneous equation He~4Z2is4 exists, where 4 is given by equation (3.7).

Using the perturbation series expansion for Fn;me, we define the expansionHeZH0CeH1Ce2H2COe3. By lemma 4.1 in Pelinovsky et al. (2005b),computations of appendix A determine the splitting of zero eigenvalues of Heas es0. The splitting of zero eigenvalues ofsHe is formulated and proved asfollows.

Proposition 3.2. LetFn;me

,

n; m

2Z

2 be a family of vortex solutions defined

by proposition 3.1. The linearized problem (3.3)(3.4) has zero eigenvalue ofalgebraic multiplicity two and geometric multiplicity one and three small pairs ofpurely imaginary eigenvalues of negative Krein signatures1 with the asymptoticapproximations

l1;2; l3;4ZG2ieCOe2; l5;6ZG4ie2COe3:The rest of the spectrum is bounded away from the origin ase/0 and it is locatedon the imaginary axis ofl.

Proof. We supplement the general proof of lemma 4.2 in Pelinovsky et al.(2005b) with the explicit perturbation series expansions for small eigenvalues of

the linear eigenvalue problem (3.5),4Z4

0Ce4

1Ce

24

2COe3; lZ el1Ce2l2COe3; 3:8

where

40Z

X4jZ1

cjej; 41Z

l1

2

X4jZ1

cjejC41inhom;

and the solution 41inhomZKH0K1H140ZKH140 is uniquely defined on the

set S1, where S1 is the set of adjacent nodes to the set of S0. At the second-order perturbation theory, the problem is written in the form

H042CH141CH240Z il1s41C il2s40: 3:9Projecting the problem to the kernel of H0, we find the reduced eigenvalueproblem

M2cZ 12l21c; 3:10

where cZc1; c2; c3; c4T and M2 is computed in appendix A. Therefore, twonegative eigenvalues g1;2 of the Jacobian matrix e

2M2 generate two pairs ofimaginary eigenvalues of negative Krein signatures1 in the linear eigenvalueproblem by virtue of the relation lZG

ffiffiffiffiffiffi2g

p. The same computation is then

extended up to the fourth order, where it is found that the negative eigenvalue g3

1 A simple eigenvalue of the linear eigenvalue problem (3.5) is said to have the negative Kreinsignature if the quadratic form for the associated eigenvector He4;4 is negative.





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of the extended matrix e2M2Ce4M4 determines the third pair of purelyimaginary eigenvalues by virtue of the same relation lZG

ffiffiffiffiffiffi2g

p. &

We note that the count of eigenvalues of negative Krein signaturescorresponds to the closure theorem for negative index of He (see Pelinovskyet al. (2005a) for details). There are four negative eigenvalues of H0 for thelimiting solution (3.2) and three more small negative eigenvalues occur for es0.

The total number of negative eigenvalues is reduced by the gauge symmetryconstraint, such that six negative eigenvalues in a constrained subspace matchthree pairs of imaginary eigenvalues with negative Krein signature.

The asymptotic approximations of eigenvalues l are plotted in figure 2 bydashed lines. The numerical computations of the same eigenvalues (up to theprescribed numerical accuracy) versus e are shown by solid lines. The numericalaccuracy is excellent for e%0:1 but it becomes worse for eO0:2 especially for thesmallest eigenvalue. The astute reader will, in fact, observe that this is a generaltrend in what follows (see figures below), i.e. higher-order predictions may bemore sensitive (as may be expected due to their higher-order corrections) tovariations of e as e grows. All three pairs of purely imaginary eigenvaluesbifurcate into complex domains when they collide to other eigenvalues of thestability problem (e.g. with eigenvalues of positive Krein signatures or with thespectral band). The first collision is numerically detected to occur at ez0:395.

4. Vector vortex crosses for 0! b! 1 and bO 1

In order to consider the coupled vortex configurations in the non-degenerate casebs1, we extend computations of appendix A to the solution of the couplednonlinear difference equations (2.4) and (2.5). We report here computations for

two related problems: (i) bifurcations of small eigenvalues of the linearizedJacobian matrix near the zero eigenvalue and (ii) bifurcations of smalleigenvalues of the linearized stability problem near the origin. Because of

0

0.5

1.0(a)

(b)

li

0.1 0.2 0.3 0.40

0.1

0.2

lr

Figure 2. Eigenvalues of the scalar vortex cross versus e. (a) Imaginary part and (b) real part of the

relevant eigenvalues. The solid lines display the numerical results, while the dashed onescorrespond to the asymptotic approximations.





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the computational complexity of the analytical approximations, we shallcomplement the analytical results of the second-order LS reductions with thesymbolic computational results of the fourth-order LS reductions.

Similarly to the scalar case, the linearized stability problem for the two-component system takes the matrix-vector form

He4Z ils4; 4:1where He is the linearized Jacobian matrix for the systems (2.4) and (2.5), s isa diagonal matrix of (1,K1, 1,K1) and 4 is an eigenvector consisting of fourelements of the perturbation vector at each node n; m2Z2. The diagonal blockof the matrix He at each node n; m2Z2 takes the form

1K2jfn;mj2Kbjjn;mj2 Kf2n;m Kbfn;mjn;m Kbfn;mjn;mKf

2n;m 1K2jfn;mj2Kbjjn;mj2 Kbfn;mjn;m Kbfn;mjn;m

Kbfn;mjn;m Kbfn;mjn;m uKbjfn;mj2K2jjn;mj2 Kj2n;mKbfn;mjn;m Kbfn;mjn;m Kj

2n;m uKbjfn;mj2K2jjn;mj2

0BBBBBB@

1CCCCCCA

:

The non-diagonal blocks ofHe come from the difference operators in the right-hand-side of the systems (2.4) and (2.5).

(a) Bifurcations of zero eigenvalues of the linearized Jacobian matrix

We extend the perturbation series expansions (A 1) to the two-component case,

fn;me

ZX

N

kZ0

ekf

kn;m; jn;m

e

ZX

N

kZ0

ekj

kn;m;

4:2

where the zero-order solution in the anti-continuum limit is given by equation (2.7)and parameters (a, b) are given in equation (2.9). The first-order corrections arefound from the uncoupled system of equations, similarly to the scalar case,

f1n;mZ

0; n; m2S0;

aX1

l

eiql; n; m2S1;

0;n; m

;S0gS1;

j1n;mZ

0; n; m2S0;

uK1bX1

l

einl; n; m2S1;

0;n; m

;S0gS1;

8>>>>>>>:

8>>>>>>>:4:3

whereP1

l is defined in (A 3). The second-order corrections are found in the form

f2n;mZ

s2

j eiqj; n;m2S0;

0; n;m2S1;

a

X2

l

eiql; n;m2S2;

0; n;m;S0gS1gS2;

j2n;mZ

r2

j einj; n;m2S0;

0; n;m2S1;

uK2b

X2

l

einl; n;m2S2;

0; n;m;S0gS1gS2;

8>>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>:4:4





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whereP2

l is defined in (A 5). The real parameters s2j ; r2j satisfy aninhomogeneous system

K2as2j Cbbr2j Z 4C2 cosqjC1KqjC2 cosqjK1KqjCcosqjC2Kqj; 4:5

K2bas2j Cbr2j Z 4C2 cosnjC1KnjC2 cosnjK1KnjCcosnjC2Knj: 4:6When bs1, the inhomogeneous system (4.5) and (4.6) has a unique solution.Second-order corrections to the bifurcation equations are uncoupled and have theform

g2

j Z 2 sinqjKqjC1C2 sinqjKqjK1CsinqjKqjC2;h

2j Z 2 sinnjKnjC1C2 sinnjKnjK1CsinnjKnjC2;

where a suitable normalization of g2

j and h2

j is made. As a result, the Jacobian

matrix computed from derivatives of g2j ; h2j T in qi; ni is block-diagonal asdiagM2;M2, where M2 is defined in appendix A. By lemma 4.1 in Pelinovskyetal.(2005b), non-zero eigenvalues of diagM2;M2 determine small eigenvalues ofthe linearized Jacobian matrix He,

g1;2;3;4ZK2e2COe4:

Two zero eigenvalues of diagM2;M2 split into two non-zero eigenvalues in thefourth-order LS reductions, while two other zero eigenvalues of diagM2;M2persist beyond all orders due to the gauge invariance of each component in coupled

DNLS equations (2.1) and (2.2). Indeed, the kernel ofHe for es0 includes at leasttwo eigenfunctions,

4n;mZ

fn;m

Kfn;m

0

0

0BBBB@

1CCCCA;

0

0

jn;m

Kjn;m

0BBBB@

1CCCCA

8>>>>>>>:

9>>>>=>>>>;

; n; m2Z2: 4:7

In order to compute the small non-zero eigenvalues of the linearized Jacobianmatrix

He

, we use the symbolic computation package based on Wolframs

MATHEMATICA. The projection to the eigenspace of diagM2; M2 spanned byeigenvectors p2; 04T and 04;p2T, where p2Z K1; 1;K1; 1 and 04Z0; 0; 0; 0,leads to the reduced eigenvalue problem (for uZ1),

K8

1Cba1Gba2Z ~ga1;

K8

1CbGba1Ca2Z ~ga2;

where

a1;a2

are coordinates of the projections, ~gZ lime/0e

K4g and the upper/

lower signs refer to the two coupled vortices (1,G1). It is clear that theeigenvalues of the reduced eigenvalue problem are the same for either signand they define two small eigenvalues of the linearized Jacobian matrix He





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(for uZ1),

g5ZK8e4COe6; g6ZK

81Kb1Cb e

4COe6:

(b ) Bifurcations of zero eigenvalues of the linearized stability problemWe consider the eigenvalue problem (4.1) in the limit of small e. Let

HeZH0CeH1Ce2H2COe3. The set of eigenvectors of H0ejZ0 andH0fjZ0 takes the form

ejZ i

eiqj

KeKiqj

0

0

0BBBBB@

1CCCCCA

; fjZ i

0

0

einj

Ke

Kinj

0BBBBB@

1CCCCCA

: 4:8

The corresponding set of generalized eigenvectors of H0ejZ2isej and H0^fjZ2isfj takes the form

ejZ

ACeiqj

ACeKiqj

BCeinj

BCeKinj

0BBBBB@

1CCCCCA; fjZ

AKeiqj

AKeKiqj

BKeinj

BKeKinj

0BBBBB@

1CCCCCA; 4:9

where

ACZ1

a21Kb2 ; BCZAKZKb

ab1Kb2 ; BKZ1

b21Kb2 ;Bifurcations of zero eigenvalues of the linear eigenvalue problem (4.1) can becomputed with the extended perturbation series expansions (4.2) for fn;me andjn;me and extended perturbation series (3.8) for 4 and l, where

40Z

X4jZ1

cjejCX4jZ1

djfj; 41Z

l1

2

X4jZ1

cjejCl1

2

X4jZ1

dj^fjC41inhom;

and4

1inhom

ZK

H0

K1

H1

4

0

ZK

H1

4

0

is uniquely defined on the set S1

. Atthe second-order perturbation theory, we have the same problem (3.9), fromwhich we derive the reduced eigenvalue problem,

M2cZ 12l21ACcCAKd; 4:10

M2dZ 12l21BCcCBKd; 4:11where cZc1; c2; c3; c4T, dZ d1; d2; d3; d4T and M2 is the same as in the scalarcase. Let g1Z1=2l21. The reduced eigenvalue problems (4.10) and (4.11) hasfour zero roots for g1 and two double-degenerate non-zero roots for g1, given

from the quadratic equation

g1C2a2g1C2b2Z 4a2b2b2: 4:12





12/25

If uZ1, such that a2Zb2Z1=1Cb, then the two non-zero roots for g1 arefound explicitly,

gGZK21Hjbj

1Cb:

By using the relation l1ZGffiffiffiffiffiffiffiffi

2g1p , we have just proved that the linear eigenvalueproblem (4.1) in the case uZ1 and 0!b!1 has four small pairs of purelyimaginary eigenvalues with asymptotic approximations,

l1;2; l3;4ZG2ieCOe2; l5;6; l7;8ZG2ieffiffiffiffiffiffiffiffiffiffiffiffi

1Kb

1Cb

sCOe2:

Two pairs of eigenvalues l5;6 and l7;8 become pairs of real eigenvalues in the casebO1. Two pairs of zero eigenvalues of the reduced eigenvalue problems (4.10)and (4.11) split at the fourth-order LS reductions as pairs of non-zero eigenvaluesl9;10 and l11;12. Two other pairs of zero eigenvalues persist beyond all orders for

es0, since the geometric kernel includes two explicit solutions (4.7) and thereexists a two-parameter solution of the inhomogeneous equation He~4Z2is4,where 4 is given by equation (4.7). In order to find the small non-zero pairs ofeigenvalues, we apply again the symbolic computation package based onWolframs MATHEMATICA. The projection to the eigenspace of diagM2;M2spanned by eigenvectors p2; 04T and 04;p2T for l1Z0 leads to the reducedeigenvalue problem (for uZ1),

K8

1Cba1Gba2Z

1

21Kb l22a1Kba2;

K81Cb

Gba1Ca2Z 121Kb l22Kba1Ca2;

where a1;a2 are coordinates of the projections and the upper/lower signs referto the two coupled vortices 1;G1. The eigenvalues of the reduced eigenvalueproblem differs between the double-charge vortex (1, 1) and the hidden-chargevortex (1,K1). For the double-charge vortex, the two pairs of small eigenvaluesof the linearized stability problem are purely imaginary for any b:

1; 1 : l9;10ZG4ie2COe3; l11;12ZG4i1Kb

1Cb e2COe3:

For the hidden-charge vortex, the two pairs of small eigenvalues of the linearizedstability problem are purely imaginary for 0!b!1 and real for bO1:

1;K1 : l9;10; l11;12ZG4iffiffiffiffiffiffiffiffiffiffiffiffi

1Kb

1Cb

se

2COe3:

We can specify precisely how many purely imaginary eigenvalues of thelinearized stability problem (4.1) have negative Krein signature. When 0!b!1,

there are eight negative eigenvalues ofH0 for the limiting solution (2.7) and sixmore small negative eigenvalues occur for es0. The total number of negative

eigenvalues is reduced by two gauge symmetry constraints, such that 12 negativeeigenvalues in a constrained subspace match six pairs of imaginary eigenvalueswith negative Krein signature. When bO1, there are four negative eigenvalues of





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H0 for the limiting solution (2.7) and five more small negative eigenvalues occurfor es0. The total number of negative eigenvalues is reduced by one2, such thateight negative eigenvalues in a constrained subspace match two real eigenvaluesand three pairs of imaginary eigenvalues with negative Krein signature for thedouble-charge vortex and four real eigenvalues and two pairs of imaginaryeigenvalues with negative Krein signature for the hidden-charge vortex.Therefore, the last pair of purely imaginary eigenvalues l11;12 for the double-charge vortex has positive Krein signature for bO1.

We obtain numerically small eigenvalues l for small values ofe and uZ

1. Theresults are shown in figure 3 for bZ2=3 and in figure 4 for bZ2. The left plotcorresponds to the vortex pair (1, 1), while the right plot corresponds to thevortex pair (1,K1). We note that the degeneracy of the pairs l1;2Zl3;4 andl5;6Zl7;8 is preserved for the case (1,K1), such that each bold curve is double.The degeneracy of these eigenvalues is broken for the case (1, 1) and it is alsobroken for the pair l9;10sl11;12 for the case (1,K1).

In the case of bZ2=3, shown in figure 3, all six pairs of neutrally stableeigenvalues bifurcate to the complex plane for larger values of e due to theHamiltonianHopf (HH) bifurcation. The first HH bifurcation happens earlier forthe case (1, 1) at ez0:395, due to the broken degeneracy between the two pairsof eigenvalues l1;2 and l3;4. For the case (1,K1), the first HH bifurcationoccurs at ez0:495, i.e. the hidden-charge vortex has a larger stability window for0!b!1 (a similar observation is reported for continuous systems inDesyatnikov et al. (2005) and Ye et al. (2004)).

In the case ofbZ2, shown in figure 4, both cases (1, 1) and (1,K1) are alwaysunstable due to the pairs of eigenvalues l5;6 and l7;8. There are also additionalobservations. In the case (1, 1), the pairs of double real eigenvalues in thesecond-order LS reductions l5;6 and l7;8 split as a quartet of complex eigenvalues,similarly to our computations in Pelinovsky et al. (2005b). Real and imaginary

0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

(a) (b)

li

0.1 0.2 0.3 0.4

Figure 3. Eigenvalues of the vector vortex cross with uZ1 and bZ2=3 versus e. (a) (1, 1) and (b)(1,K1). The solid lines show the numerical results, while the dashed lines show the asymptoticapproximations. Bold curves correspond to double eigenvalues (that remain indistinguishable

within the parametric window examined herein). A good agreement is observed for e!0:1.

2 When b is increased from b!1 to bO1, the Hessian matrix related to two gauge symmetryconstraints loses one positive eigenvalue that passes through zero at bZ1 to the negativeeigenvalue for bO1 (Pelinovsky & Kivshar 2000).





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parts of the quartet of complex eigenvalues are shown in figure 4aby bold curves.Only three HH bifurcations out of four pairs of purely imaginary eigenvaluesoccur for larger values ofe. In the case (1,K1), two more pairs of real eigenvaluesoccur such that the hidden-charge vortex is more unstable compared to thedouble-charge vortex for bO1. Only two HH bifurcations occur for large valuesof e.

5. Vector vortex cross for bZ

1

The coupled DNLS system in the symmetric case bZ1 is referred to as thediscrete Manakov system. In the case bZ1, the existence domain of the coupledvortex configurations shrinks to the line uZ1. The zero-order solution in theanti-continuum limit is given by equation (2.7), where parameters (a, b) aregiven by equation (2.11). The second-order solution of the linear inhomogeneoussystems (4.5) and (4.6) with a singular matrix exists provided that the values ofqj and nj are defined by equation (2.12). The arbitrary parameter in the second-

order solution s2

j and r2

j renormalizes the arbitrary parameter d in the

representation (2.11).When bZuZ1, the existence problem (2.4)(2.5) is symmetric with respect tocomponents fn;m;jn;m such that the systems (2.4) and (2.5) can be reduced to thescalar difference equation (3.1) with the two independent transformations

1; 1 : fn;mZ cos dFn;m; jn;mZ sin dFn;m;1;K1 : fn;mZ cos dFn;m; jn;mZ sin d Fn;m:

The existence result for the scalar vortex cross is formulated in proposition 3.1. Wewill need the following non-degeneracy condition for the scalar vortex cross:

Xn;m2Z2

jFn;mj20@ 1A2s X

n;m2Z2F2n;m

0@ 1A Xn;m2Z2

F2n;m0@ 1A: 5:1

0

0.5

(a) (b)

li

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.40

0.5

lr

Figure 4. Eigenvalues of the vector vortex cross with uZ1 and bZ2 versus e. (a) (1, 1) and (b)(1,K1). The solid lines show the numerical results, while the dashed lines show the asymptoticapproximations. Bold curves in (a) correspond to the real and imaginary parts of complex

eigenvalues, while bolded curves in (b) correspond to double eigenvalues.





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It is clear from the limiting solution (2.7) that the constraint (5.1) is satisfied forsmall e. The stability problem (4.1) is different between the cases (1, 1) and (1,K1).

(a) Eigenvalues of the (1, 1) vortex cross

In this case, the stability problem (4.1) is block-diagonalized under thefollowing transformation of the four components of the vector f on the latticenode n; m2Z2,

1; 1 :

an;m

bn;m

cCn;m

cKn;m

0BBBB@

1CCCCAZ

cos d 0 sin d 0

0 cos d 0 sin d

Ksin d 0 cos d 0

0 Ksin d 0 cos d

0BBBB@

1CCCCA

41

42

43

44

0BBB@

1CCCA

n;m

:

The components an;m; bn;m satisfy the linear eigenvalue problem for scalarvortices (3.3) and (3.4). The components cCn;m; cKn;m satisfy two uncoupledself-adjoint eigenvalue problems,

1KjFn;mj2cGn;mKecGnC1;mCcGnK1;mCcGn;mC1CcGn;mK1ZGilcGn;m: 5:2

Using the result of proposition 3.2 and equivalent computations for theuncoupled self-adjoint problems (5.2), we prove the following result.

Proposition 5.1. LetFn;m

e,n; m

2Z

2 be a family of vortex solutions definedby proposition 3.1. The linearized problem (4.1) in the case bZ1 for the (1, 1)vortex cross has a zero eigenvalue of algebraic multiplicity six and geometricmultiplicity five and five small pairs of purely imaginary eigenvalues givenasymptotically by

l1;2; l3;4ZG2ieCOe2; l5;6ZG2ie2COe3;l7;8ZG6ie

2COe3; l9;10ZG4ie2COe3:

The rest of the spectrum is bounded away the origin ase/0 and it is located on

the imaginary axis ofl.Proof. It remains to study bifurcations of zero eigenvalues in the self-adjoint

problem (5.2) as es0. Let us define the perturbation series for the problem (5.2),

cGZ c0Cec

1Ce

2c2COe3; lZGie2l2COe3:

The zero-order solution is spanned by unit vectors ej at the jth component thatcorrespond to the node n; m2S0,

c0ZX

4

jZ1

ajej:





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The first-order correction c1 takes the form

c1n;mZ

0; n; m2S0;

X1

l

al; n; m

2

S

1;

0; n; m;S0gS1;

8

>>>>>>>:where the sum

P1l is defined in equation (A 3). At the second-order in e, we find

a set of non-trivial equations at the nodes n; m2S0,ajCajC2C2ajC1CajK1Z l2aj; jZ 1; 2; 3; 4:

The reduced eigenvalue problem has a double zero eigenvalue and twonon-zero eigenvalues K2 and 6. Two zero eigenvalues of the problem (5.2)

persist at all orders ofe, because of the exact solutions: c

G

n;mZF

n;m andcGn;mZ Fn;m. &

We note that the pairs of eigenvalues l1;2, l3;4 and l9;10 continue theeigenvalues of the vortex cross (1, 1) from bs1 to bZ1. The pairs of eigenvaluesl5;6 and l7;8 match with the zero values of the Oe corrections to thecorresponding eigenvalues of the vortex cross (1, 1) for bs1. Finally, the pair ofnon-zero eigenvalues l11;12 for bs1 is forced to remain at the origin for bZ1 dueto the polarizationrotation symmetry.

We can now specify how many purely imaginary eigenvalues l have negativeKrein signature. When bZ1, there are 4 negative and 12 zero eigenvalues of

H0 for the limiting solution (2.7). Out of the 12 zero eigenvalues, three small

negative eigenvalues bifurcate in the subspace for components an;m; bn;m, twosmall positive and two small negative eigenvalues bifurcate in the subspace forcomponents cCn;m; cKn;m and five eigenvalues remain at zero as es0. The totalnumber of negative eigenvalues is reduced by one symmetry constraint3, suchthat eight negative eigenvalues in a constrained subspace match four pairs ofimaginary eigenvalues with negative Krein signature. The only pair of purelyimaginary eigenvalues with positive Krein signature is the pair l5;6 that isrelated to the two small positive eigenvalues in the subspace for componentscCn;m; cKn;m.

(b ) Eigenvalues of the (1,K1) vortex cross

Since the stability problem (4.1) has no block-diagonalization for the (1,K1)vortex cross, the results of the second-order LS reductions give only two pairs ofpurely imaginary eigenvalues l1;2 and l3;4. We shall study the eigenvalues of thefourth-order LS reduction by using the symbolic computation package based onWolframs MATHEMATICA. In order to prepare for symbolic computations, we notethat the eigenvalues of He in the case (1,K1) are exactly the same aseigenvalues of

He

in the case (1, 1), due to the equivalent transformation of the

3 The Hessian matrix related to two gauge symmetry constraints has a zero eigenvalue for bZ1,while only positive eigenvalues are counted in a reduction of the negative index of He.





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vector 4 in the eigenvalue problem He4Zg4:

1;K1

:

an;m

bn;m

cC

n;m

cKn;m

0

BBBB@

1

CCCCAZcos d 0 0 sin d

0 cos d sin d 0

0 Ksin d cos d 0Ksin d 0 0 cos d

0

BBBB@

1

CCCCA41

42

4344

0BBB@

1CCCA

n;m

:

As a result of this transformation, we immediately find the five-dimensionalkernel of He for es0, which can be spanned as follows:

4n;mZ

cos d Fn;m

Kcos d Fn;m

Ksin d Fn;m

sin d Fn;m

0BBBBB@

1CCCCCA

;

Ksin d Fn;m

0

0

cos d Fn;m

0BBBB@

1CCCCA

;

0

Ksin d Fn;m

cos d Fn;m

0

0BBBB@

1CCCCA

;

Ksin d Fn;m

0

0

cos d Fn;m

0BBBB@

1CCCCA

;

0

Ksin d Fn;m

cos d Fn;m

0

0BBBB@

1CCCCA

8>>>>>>>>>:

9>>>>>=

>>>>>;;

5:3for n; m2Z2. The algebraic multiplicity of zero eigenvalue for es0 is definedby the solution of the inhomogeneous equation He~4Z2is4, which is equivalentto the projection equationsX

n;m2Z2h4j; s4iZ 0; jZ 1; 2; 3; 4; 5;

where 4 is spanned by five eigenvectors 4j in the decomposition (4.7). Solvingthis system of linear equations, we have found under the non-degeneracy

condition (5.1) that there is a one-parameter solution of the inhomogeneoussystem for dsp=4 and a three-parameter solution for dZp=4. Thus, the zeroeigenvalue has algebraic multiplicity six for dsp=4 and eight for dZp=4.

In the limit eZ0, when H0ZH0, we construct explicitly three sets of linearlyindependent eigenvectors ofH0,

ejZ i

cos d eiqj

Kcos d eKiqj

sin d eKiqj

Ksin d eiqj

0BBBBB@

1CCCCCA

; fCj Z i

cos d eiqj

Kcos d eKiqj

Ksin d eKiqj

sin d eiqj

0BBBBB@

1CCCCCA

; fKj Z

sin d eiqj

sin d eKiqj

Kcos d eKiqj

Kcos d eiqj

0BBBBB@

1CCCCCA

: 5:4

Only the set of eigenvectors ej generates the set of generalized eigenvectors of theproblem H0ejZ2isej, where

ejZ

cos d eiqj

cos d eKiqj

sin d eKiqj

sin d eiqj

0BBBBB@

1CCCCCA

: 5:5

Thus, the zero eigenvalue ofH0 has algebraic multiplicity sixteenand geometricmultiplicity twelve. Two pairs of purely imaginary eigenvalues of negative Krein





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signatures bifurcate at the second-order LS reductions as

l1;2; l3;4ZG2ieCOe2:In order to study bifurcations of non-zero eigenvalues at the fourth-order LSreductions, we consider the extended perturbation series (3.8) for 4 and l with

l1Z0 and

40Z

X4jZ1

cjejCX4jZ1

dCj fCj C

X4jZ1

dKjfKj : 5:6

Performing computations symbolically, we have 12 homogeneous equations at the

order of Oe2 for 12 variables cj; dCj ; dKj , jZ1; 2; 3; 4, which can be convertedand simplified to the following determinant equation:

g22C41C4 cos 4dg2C36Z 0;

where g2Z1=2l22. By using the inverse relation l2ZGffiffiffiffiffiffiffiffi2g2p

and finding the roots

for g2 explicitly, we obtain four small pairs of eigenvalues with asymptoticapproximations,

l5;6ZG2ie2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1C4 cos 4dKffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8cos 4dCcos 8dpq COe3;l7;8ZG2ie

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1C4 cos 4dCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8cos 4dCcos 8dpq COe3:When dZ0 or dZp=2, we obtain the same pairs of purely imaginary eigenvaluesas in the case (1, 1) (see proposition 5.1). When dZp=4, we obtain two degeneratepairs of real eigenvalues,

l5;6; l7;8ZG2 ffiffiffi3p e2COe3

:

The instability domain is found analytically from the condition thatcomplex-valued roots for g2 coalesce and become a double negative root.This happens when cos4dCcos8dZ0, which is solved on the intervald2 0;p=2 at dZp=12 and dZ5p=124. Thus, the instability domain ofthe (1,K1) vortex cross in the case bZ1 is bounded by the intervald2p=12; 5p=12.

In order to capture the remaining pair of non-zero eigenvalues l9;10, we shallreorder the perturbation series expansions and to move the last two sums in thedecomposition (5.6) to the order of Oe2, while the coefficients of the vectorcZc1; c2; c3; c4

T

should be projected to the vector p2ZK1; 1;K1; 1 of thekernel ofM2, such that cZx1p2. Performing computations symbolically, we havetwelve homogeneous equations at the order of Oe4 for eight variables in thevectors dC and dKj and the coordinate x1. The homogeneous system is satisfied

with the choice dCZ0 and dKZx2p2, where x2 is another coordinate. Thecoordinates x1; x2 solve a homogeneous system with the determinant equationl22ZK16 cos

22d. Therefore, a small pair of purely imaginary eigenvalues ofnegative Krein signatures has the asymptotic approximation

l9;10ZG4ie2cos2dCOe3:

4 Another solution exists at dZp=4, but it corresponds to the case when complex-valued rootscoalesce and become a double positive root for g2.





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When dZ0 and dZp=2, the pair l9;10 matches to that in the case (1, 1) (seeproposition 5.1). When dZp=4, the pair remains at the origin as it follows from thestudy of algebraic multiplicity of zero eigenvalue. According to the count ofnegative eigenvalues, the total number of negative eigenvalues ofHe for small ereduced by one symmetry constraint is eight. These eigenvalues match two pairs ofimaginary eigenvalues l1;2 and l3;4 and two real positive eigenvalues l5;6 and l7;8

5.Asymptotic and numerical approximations of small eigenvalues l for small

values ofe for uZbZ1 and dZp=4 are shown in figure 5. Figure 5acorrespondsto the vortex pair (1, 1), while figure 5b corresponds to the vortex pair (1,K1).We can see that the (1, 1) vortex cross is linearly stable in the anti-continuumlimit, according to the results of proposition 5.1. On the other hand, the (1,K1)vortex cross becomes unstable because of the double pairs of real eigenvaluesl5;6Zl7;8. The other double pair of purely imaginary eigenvalues remains doublefor all eO0, such that l1;2Zl3;4. Therefore, the stability changes drastically inthe case bZ1: the (1, 1) vortex cross is stable near the anti-continuum limit whilethe (1,K1) vortex cross is linearly unstable.

6. Conclusion

We have examined analytically and numerically the existence and stability ofvortex cross configurations in the single-component and two-component DNLSequations. We have used the LyapunovSchmidt theory, to obtain thebifurcation functions and the solvability conditions that allow persistence ofsuch configurations near the anti-continuum limit. Additionally, the theorygives analytical expressions for eigenvalues of the linearized stability problemas functions of the system parameters (namely, the coupling between adjacent

0.1 0.2 0.3 0.40

0.2

0.4

0.6

(a) (b)0.8

0.1 0.2 0.3 0.4

li

li

lr

0

0.5

0

0.2

0.4

Figure 5. Eigenvalues of the vector vortex cross with uZbZ1 and dZp=4 versus e. (a) (1, 1) and(b) (1,K1). The solid lines show the full numerical results, while the dashed lines show the

asymptotic approximations. Bold curves show double eigenvalues.

5 Eigenvalues l5,6 and l7,8 are real only in the case dZp=4. For other values ofd, these eigenvaluesare either complex-valued or purely imaginary. The count is not affected, since two pairs of real

eigenvalues are equivalent to four complex eigenvalues which may coalesce due to the inverseHamiltonHopf bifurcation to two pairs of purely imaginary eigenvalues with positive and negativeKrein signatures.





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lattice nodes eR0 and the coupling between the two components bR0). Webelieve that similarly to what was experimentally shown for the scalar vortex-cross configuration in Neshev et al. (2004) and Fleischer et al. (2004), thestable vector vortex-cross configurations can be observed experimentally eitherin waveguide array experiments (Meier et al. 2003) or in atomic physics

experiments of coupled hyperfine states of BECs in deep optical lattices(Myatt et al. 1997). While in the former context, specific values of thenonlinear coupling b are selected by the nature of the physical model, in thelatter there is even the versatile capability of tuning b at will via the so-calledFeshbach resonance (Brazhnyi & Konotop 2004) and observing a wide range ofphenomena presented herein.

P.G.K. is supported by NSF through the grants DMS-0204585, DMS-CAREER and DMS-0505663.D.E.P. is supported by NSERC and PREA grants.

Appendix A. Continuation of the single-component vortex cross

We apply the algorithm of LS reductions (see Pelinovsky et al. (2005b) fordetails) and compute the first few terms of the perturbation series expansions,

Fn;meZXNkZ0

ekF

kn;m: A 1

The zero-order solution F0n;m is given by (3.2). The first-order correction is

obtained in the explicit form

F1n;mZ

0;

n; m

2S0;

X1l

eiql; n; m2S1;0; n; m;S0gS1;

8>>>>>>>:

A 2

where S1 is the set of adjacent nodes to the set S0 andP1

l eiqlis a schematic

notation for the following solution:

X1

l

eiqlZ

eiq1Ceiq2Ceiq3Ceiq4 ; n; mZ 0; 0;eiqjCeqjC1 ; n; mZ fK1;K1; 1;K1; 1; 1; K1; 1g;eiqj;

n; m

Z

fK2; 0

;

0;K2

;

2; 0

;

0; 2

g:

8>:

A 3

The index jenumerates nodes in the set S0 that are adjacent to the nodes in theset S1 listed in the figured brackets of (A 3). No non-trivial bifurcation equationsarise at the first-order reductions, i.e. the first-order correction to the bifurcationfunction g1q is zero, where qZ q1; q2; q3; q4 and notations ofPelinovsky et al.(2005b) are used. The second-order correction is found in the form

F2n;mZ

s2

j eiqj; n; m2S0;

0; n; m2S1;

X2l

eiql

; n; m2S2

;0; n; m;S0gS1gS2;

8>>>>>>>>>>>>>:

A 4





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where

K2s2

j Z 4C2 cosqjC1KqjC2 cosqjK1KqjCcosqjC2Kqj:The set S2 contains outward adjacent nodes to the set S1nf0; 0g and

P2l e

iql

is a schematic notation for the following solution:

X2l

eiqlZ

2eiqjCeiqjC1 ; n; mZ fK2;K1; 1;K2; 2; 1; K1; 2g;eiqjC2eqjC1 ; n; mZ fK1;K2; 2;K1; 1; 2; K2; 1g;eiqj; n; mZ fK3; 0; 0;K3; 3; 0; 0; 3g:

8>>>: A 5

The second-order corrections to the bifurcation function take the form

g2

j Z 2 sinqjKqjC1C2 sinqjKqjK1CsinqjKqjC2; jZ 1; 2; 3; 4: A 6The bifurcation equations g2qZ0 are satisfied with the one-parameter familyof asymmetric vortices,

q1Z 0; q2Z q; q3Zp; q4ZpCq; A 7where q20;p. When qZp=2, the family (A 7) reduces to the vortex crossconfiguration (2.12). The Jacobian matrix M2 of the second-order bifurcationfunction g2q is obtained by differentiation of g2 in q. At the family ofasymmetric vortices (A 7), the Jacobian matrix M2 takes the form

M2Z

K1 K2 cos q 1 2 cos q

K2 cos q K1 2 cos q 1

1 2 cos q K1 K2 cos q2 cos q 1 K2 cos q K1

0

BBBB@

1

CCCCA:

It has two zero eigenvalues and two non-zero eigenvaluesK2G 4 cos q. In the caseof the vortex cross (qZp=2), it has two zero eigenvalues and two negativeeigenvalues K2. The third-order correction satisfies the inhomogeneousequation,

1K2jF0n;mj2F3n;mKF02n;mF3n;mZF2nC1;mCF2nK1;mCF2n;mC1CF2n;mK1CjF1n;mj2F1n;m;

where we have shortened nonlinear terms, since F0n;mF

1n;mZF

1n;mF

2n;mZ0 for all

n; m2Z2. The third-order correction is found in the form

F3n;mZ

0; n; m2S0;

jF1n;mj2F1n;mCX1

l

s2l e

iqlCX1;3

l

eiql; n; m2S1;

0; n; m2S2;

X3

l

eiql;n; m

2S3;

0; n; m;S0gS1gS2gS3;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>:

A 8





22/25

where the sumP1

l s2l e

iql is defined similarly to the sum (A 3), the sumP3

l eiql

is not used for further computations and the sumP1;3

l eiql is defined as follows:

X1;3

l

eiqlZ3eiqjC3eqjC1 ; n; mZ fK1;K1; 1;K1; 1; 1; K1; 1g;

5eiqjCe

iqjC1Ce

iqjK1

; n; mZ fK2; 0; 0;K2; 2; 0; 0; 2g;( A 9

No non-trivial bifurcation equations arise at the third-order reductions, i.e.g3qZ0. The fourth-order correction satisfies the inhomogeneous equation,

1K2jF0n;mj2F4n;mKF02n;mF4n;mZ 2jF2n;mj2F0n;mCF22n;mF0n;mCF3nC1;mCF3nK1;m

CF3n;mC1CF

3n;mK1:

Solving the inhomogeneous equation for the third-order corrections, we obtainthe bifurcation equations at the fourth order of LS reductions in the form

g4

j Z4C2cosqjC2KqjC1C2cosqjKqjC1CcosqjK1KqjC1sinqjC1Kqj

C4C2cosqjK2KqjK1C2cosqjKqjK1CcosqjC1KqjK1sinqjK1Kqj

C1

24C2cosqjK1KqjK2C2cosqjC1KqjC2CcosqjKqjC2sinqjC2Kqj

C1

24C2cosqjC1KqjC2C2cosqjK1KqjK2CcosqjKqjK2sinqjK2Kqj

C21CcosqjC1KqjsinqjKqjC1C21CcosqjK1KqjsinqjKqjK1C22Ccosq2Kq1Ccosq3Kq1Ccosq4Kq1Ccosq3Kq2

Ccosq4Kq2Ccosq4Kq3sinqjKqjC1CsinqjKqjK1CsinqjKqjC2

C4sinqjKqjC1C4sinqjKqjK1:

For the asymmetric vortex, we have

g4j Z K1j2 sin2q; jZ 1; 2; 3; 4:The Jacobian matrix M2 has two zero eigenvalues with orthogonal eigenvectors,

p1Z

1

1

1

1

0BBBB@

1CCCCA; p2Z

K1

1

K1

1

0BBBB@

1CCCCA:

It is clear that the vector g4Z2 sin2qp2 is not orthogonal to the eigenvector

p2 of the kernel ofM2, unless qZf0;p=2;pg. By proposition 2.10 in Pelinovskyet al. (2005b), the family of asymmetric vortices (A 7) terminates at the





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fourth-order reduction. The exceptional cases include discrete solitons for qZf0;pg and the vortex cross at qZp=2. In order to consider persistence of thevortex cross, we compute the Jacobian matrices M2 and M4 from the bifur-cation functions g2 and g4 explicitly,

M2Z

K1 0 1 0

0 K1 0 1

1 0 K1 0

0 1 0 K1

0BBBB@1CCCCA; M4Z

3 2K

7 22 3 2 K7

K7 2 3 2

2 K7 2 3

0BBBB@1CCCCA:

Since M4p1Z0 and M4p2s0, the zero eigenvalue of M2 with the associatedeigenvector p2 bifurcates. By proposition 2.9 in Pelinovsky et al. (2005b), thisimplies that the family of the vortex cross is continued from the anti-continuumlimit uniquely up to the rotational transformation q/qCq0p1 that correspondsto the gauge symmetry of the DNLS equation (3.1). Proposition 3.1 is henceproved.

Small eigenvalues of the linearized Jacobian matrix He are defined by anextended eigenvalue problem for the Jacobian matrices M2 and M4,

e2M2Ce4M4COe6cZgc:There exist four eigenvalues of the extended problem which admit theasymptotic approximations,

g1;2ZK2e2COe4; g3ZK8e4COe6; g4Z 0:

The eigenvalue g3 is obtained by the perturbation theory for the zero eigenvalueof M2 associated with the eigenvector p2 (orthogonal to the eigenvector p1),

lime/0

eK4g3Z

p2;M4p2p2;p2

ZK8:

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Documents

P. G. Kevrekidis and D.E. Pelinovsky- Discrete vector on-site vortices