Generalized neighbor-interaction models induced by nonlinear lattices
F. Kh. Abdullaev,1,*
Yu. V. Bludov,2,†S. V. Dmitriev,3,‡P. G. Kevrekidis,4,§and V. V. Konotop2,5,储1Instituto de Física Teórica, UNESP, Rua Pamplona, 145, Sao Paulo, Brazil 2
Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal
3
General Physics Department, Altai State Technical University, 656038 Barnaul, Russia 4
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003, USA 5Departamento de Física, Universidade de Lisboa, Campo Grande, Ed. C8, Piso 6, Lisboa 1749-016, Portugal
共Received 13 July 2007; published 23 January 2008兲
It is shown that the tight-binding approximation of the nonlinear Schrödinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibili-ties, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrödinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.
DOI:10.1103/PhysRevE.77.016604 PACS number共s兲: 05.45.Yv
I. INTRODUCTION
It is generally recognized that mapping of a nonlinear evolution problem, described by a partial differential equa-tion, into a simplified lattice, representing a set of coupled ordinary differential equations, appears to be a useful tool either for numerical 共or semianalytical in the appropriate limits兲 study of the dynamics or for bringing intuitive under-standing of the factors dominating the behavior of the sys-tems. Examples of such an approach are well known for a long time in solid state physics关1兴 共e.g., the description of an electron in a crystal in the tight-binding approximation兲, in optics关2兴 共e.g., the description of the electric field in arrays of waveguides 关3,4兴兲, and more recently in the theory of matter waves in optical lattices共i.e., to the mean-field theory of arrays of Bose-Einstein condensates, which are described by the order parameter, rather than by the density matrix; this rules out the problem of ordering of field operators highly relevant to the case where each array contains only a few atoms, see, e.g.,关5,6兴 for relevant reviews兲. In all the above cases, the periodicity is typically associated with the linear properties of the system and the respective dynamics is ap-proximately described by the discrete nonlinear Schrödinger 共DNLS兲 equation. The properties, relevance, and applica-tions of the DNLS equation in diverse physical problems were analyzed in a variety of recent reviews关7,8兴. We also note in passing that there has been a wide range of recent experimental works 共and related theoretical works兲 associ-ated with this prototypical DNLS equation including, e.g., the existence of Peierls-Nabarro barriers for discrete solitons 关9兴, the modulational instability of uniform states both in optical experiments关10兴 and in BEC theory 关11兴 and
experi-ments关12兴, the examination of multicomponent DNLS lat-tices关13兴, the study of discrete surface solitons 关14兴 in one or two dimensions, and the formation of discrete vortices关15兴 in two dimensions, among many others.
On the other hand, there has recently been an increasing interest in studying nonlinear models, where the nonlinearity is also periodically modulated in space. Applications of such models extend from the propagation of electromagnetic waves in stratified media 关16,17兴 to condensates of bosons 关18兴 and condensates of boson-fermion mixtures 关19兴 in op-tical lattices. It turned out that the spatially dependent non-linearity may dramatically change properties of the system, in particular, the regions of existence of coherent localized structures and especially their corresponding stability properties.
Natural questions that arise in this context concern the mapping of the respective evolution equation into nonlinear lattice dynamics and the description of the existence, stabil-ity, and dynamical properties of such a lattice. In the present paper we consider both of these issues. While the first issue is technical and can be straightforwardly addressed by means of the Wannier function expansion, as it was suggested in 关20兴, the study of the properties of the emerging lattices is a much richer problem in the present setting; its richness stems from the fact that the spatially dependent nonlinearity gives rise to complex nonlinear intersite interactions. Such forms of nonlinearity as the ones extracted below can significantly change the dynamical properties of the discrete system as it was shown in earlier research devoted to the spin waves in magnetic systems 关21兴, to electromagnetic waves in wave-guide arrays关22,23兴, and to arrays of Bose-Einstein conden-sates关24兴. In the above mentioned studies of nonlinear lat-tices, however, one common feature was of crucial importance—that was the presence of dominant共or at least significant兲 on-site nonlinearity, which, e.g., in the case of a BEC loaded in an optical lattice is typically about two orders of magnitude larger than the hopping nonlinearity. In the present paper, we systematically derive and consider a far more general class of lattice evolution equations, including the cases where the on-site nonlinearity is exactly zero. *fatkh@uzsci.net †bludov@cii.fc.ul.pt ‡ dmitriev.sergey.v@gmail.com § kevrekid@math.umass.edu 储 konotop@cii.fc.ul.pt
It should be noted that such an analysis of reduced dis-crete models may prove particularly valuable in understand-ing either analytically or numerically the properties 共espe-cially the existence and stability兲 of solutions in the original partial differential equation 共PDE兲 models from which the discrete ones arise. More specifically, in the discrete models, it is not only simpler and more inexpensive to perform nu-merical computations, but it is often possible to completely classify their solutions 共see, e.g., for the prototypical ex-ample of the DNLS lattice the work of关25兴兲 and their corre-sponding stability共see, e.g., for the same lattice, the work of 关26兴兲. This allows us to infer useful conclusions for the cor-responding solutions in the original PDE.
The organization of the paper is as follows. In Sec. II we deduce the relevant lattice dynamical models with intersite nonlinearity starting with the evolution equation of the non-linear Schrödinger共NLS兲 type with a spatially periodic po-tential and periodic nonlinearity. In Sec. III, we conduct the modulational stability analysis of the derived models. In Secs. IV and V, we examine the respective dynamical prop-erties of the derived quasilinear and nonlinear models, while Sec. VI summarizes our findings and presents our conclu-sions, as well as directions of potential future interest.
II. MODEL EQUATIONS A. One-band approximation
We start with the one-dimensional NLS equation
i t = −
2
x2 +U共x兲+G共x兲兩兩
2, 共1兲
whereU共x兲 and G共x兲 are the coordinate-dependent linear and nonlinear potentials, respectively, both considered to be
-periodic functions:U共x兲=U共x+兲 and G共x兲=G共x+兲. We concentrate on the cases where the linear potential is an even function U共x兲=U共−x兲, while the nonlinearity may be either even or odd: G共x兲=G共−x兲 共hereafter =⫾1兲. It is impor-tant to point out here that the starting point of Eq.共1兲 implies the fact that we operate within the mean-field limit共relevant to temperatures very close to T = 0兲 and the superfluid regime of the condensate; see关5,6兴 for more details.
In order to map Eq.共1兲 into a lattice equation we follow 关20兴. To this end, we introduce the linear eigenvalue problem
− d 2
␣q共x兲
dx2 +U共x兲␣q共x兲 = E␣q␣q共x兲, 共2兲
where␣q共x兲 is a Bloch function, and␣艌1 and q stand for the band number and for the Bloch wavenumber in the first Brillouin zone: q苸关−1,1兴, and define the Wannier functions
wn␣共x兲 =
1
冑
2冕
−11
␣q共x兲e−inqdq, 共3兲 which constitute an orthonormal set of real and exponentially decaying functions关27兴.
We seek the solution of Eq. 共1兲 in the form of a series over the full orthonormal set of Wannier functions
共x,t兲 =
兺
n,␣
cn␣共t兲wn␣共x兲. 共4兲
For the next consideration we notice that the Wannier func-tions of the ␣th band possess either even or odd parity
w0␣共x兲=共−1兲1+␣w0␣共−x兲 and are characterized by the
prop-erty wn␣共x兲=w0␣共x−n兲.
Now we make the most crucial approximation of our model, namely, that the continuum Eq.共1兲 can be accurately described within the one-band approximation. As it was shown in关20兴, this assumption fails to describe the original continuous model when one studies dynamical processes as-sociated with the generation of the frequencies belonging to the higher bands. However, it is reasonably accurate in de-scribing static solutions 共in particular, localized modes兲 as well as their stability. Also, the lattices of generalized neigh-bor interactions derived below within the framework of the one-band approximation, are of interest in their own right, per their particularities and differences in comparison to other models of similar type; cf.关22,23兴.
We thus assume that only one band, say the␣th one, is populated. Now, substituting the expansion共4兲 in Eq. 共1兲 we arrive at the equation共see 关5,20兴 for more details兲
ic˙n␣− cn␣0␣−共cn−1,␣+ cn+1,␣兲1␣ −
兺
n1,n2,n3 cn1␣¯cn2␣cn3␣W␣␣␣␣ nn1n2n3 = 0, 共5兲 where W␣␣ 1␣2␣3 nn1n2n3 =冕
−⬁ ⬁ G共x兲wn␣共x兲wn1␣1共x兲wn2␣2共x兲wn3␣3共x兲dx 共6兲 are the nonlinear overlap integrals, and n␣ are thecoeffi-cients of the Fourier series expansion of the eigenvalueE␣q as follows: E␣q=
兺
n n␣einq, n␣= 1 2冕
−1 1 E␣qe−inqdq. 共7兲 The overbar stands for complex conjugation, and an overdot stands for the derivative with respect to time. In Eq.共5兲 we have taken into account that in a general situation, for a periodic potentialU共x兲 of rather large amplitude, the lowest bands are very narrow, and hence the Fourier coefficientsn␣ decay rapidly with increasing n 共so that 兩0␣兩Ⰷ兩1␣兩
Ⰷ兩2␣兩兲, which, in turn, allows us to neglect the coefficients
n␣ with n艌2.
For the lowest bands the Wannier functions are well lo-calized on the scale of one lattice period共and can be reason-ably well approximated by the eigenstates of the linear os-cillator兲 and thus the overlap integrals involving next-nearest neighbors 共i.e., lattice minima separated by two lattice maxima兲 are negligibly small. This allows us to drop also the terms involving W␣
1␣2␣3␣4
n1n2n3n4
with at least one pair of the upper indices satisfying 兩nj− nk兩艌2. We, however, emphasize that
it is of crucial importance to leave the nonlinear terms with hopping between the neighbor sites, which for specific choices of the nonlinear interactionsG共x兲 can be comparable
with or even stronger than the on-site nonlinearity共see be-low兲.
Now we use the symmetry of the integrals W with respect to permutations of the indices and introduce
W0= W␣␣␣␣nnnn = W␣␣␣␣0000 ,
W1= W␣␣␣␣n,n−1,n−1,n−1=W␣␣␣␣n,n,n,n−1= W␣␣␣␣1000 ,
W2= W␣␣␣␣n,n,n−1,n−1= W␣␣␣␣1100 , 共8兲 where= 1 and= −1 forG共x兲 even and odd and
Wj=
冕
−⬁ ⬁
G共x兲w1␣j 共x兲w0␣4−j共x兲dx, j = 0,1,2. 共9兲 In the case of odd nonlinearity共= −1兲 the terms W0and W2 are always equal to zero due to the fact that the integrand in Eq. 共9兲 is odd with respect to the points X=0 and X=/2, correspondingly. We thus arrive at the equation
ic˙n=0cn+1共cn−1+ cn+1兲 + W0兩cn兩2cn+ W1共兩cn−1兩2cn−1 +¯cn−1cn 2 + 2兩cn兩2cn−1+ 2兩cn兩2cn+1+ c¯n+1cn 2 +兩cn+1兩2cn+1兲 + W2共2兩cn−1兩2cn+ c¯ncn−1 2 + c¯ncn+1 2 + 2兩cn+1兩2cn兲, 共10兲
where we have dropped the zone index␣共e.g., cn,␣ is
rede-fined as cn, etc.兲
Equation 共10兲 is the main discrete model studied in the present paper. We notice that it has a Hamiltonian structure:
ic˙n=H/¯cnwith the Hamiltonian
H =
兺
n冋
0兩cn兩2+1共cn−1¯cn+ c¯n−1cn兲 + W0 2 兩cn兩 4+ W 1共兩cn−1兩2 +兩cn兩2兲共cn¯cn−1+ c¯ncn−1兲 + 2W2兩cn−1兩2兩cn兩2+ W2 2 共cn 2 c ¯n−12 + c¯n2cn−12 兲册
, 共11兲and with the standard Poisson brackets. Another integral of motion is the sum N =兺n兩cn兩2, reflecting the conservation of
the “number of atoms”共in keeping with the BEC motivation of our analysis兲 of the original Eq. 共1兲.
B. Particular cases
So far, we have established that the spatial dependence of the nonlinearity plays a critical role in determining the values of the nonlinear coefficients and thus accordingly determines the type of the discrete equation 共10兲. Now we consider a number of particular cases. First of all, however, we notice that in the case of odd nonlinearity function, i.e., for= −1, neither purely even共cn= c−n兲, nor purely odd 共cn= −c−n兲 so-lutions can exist. This follows directly from the symmetry of Eq.共10兲 关or Eq. 共1兲兴.
For illustration of the localized solutions we use the po-tential U共X兲=−3 cos共2X兲 共both for the detailed calculations of this section, and for the numerical simulations of the
fol-lowing sections兲. Also we restrict our considerations to the first band, i.e., we take␣= 1. In that case, the linear overlap coefficients for the first band are computed as 0⬇−0.839,
1⬇−0.051, and2⬇0.002 关37兴.
One can distinguish five cases as follows:
Case 1. W0= W1= W2= 0. This is a quasilinear case,
which is made possible, for example, by an odd nonlinearity 共= −1兲 of the form
G共x兲 = sin共2x兲 − 1.3706 sin共4x兲 共12兲
共obviously the number of possible realizations of this and other cases reported below is naturally unlimited兲. Our ap-proach both in this example and below is motivated by the nature of the lattice that we wish to construct 共i.e., by the type of overlap integral that we wish to preserve or elimi-nate兲. For instance, in this example, the odd nonlinearity guarantees that W0= W2= 0, while the expression of Eq.共12兲 uses one tunable parameter共the amplitude of the second har-monic兲 to achieve W1= 0.
Case 2. W0= W2= 0, W1⫽0, is achieved, e.g., by choosing
G共x兲 = 10 sin共2x兲 共13兲
共= −1兲. Now the lattice model 共10兲 is reduced to
ic˙n=0cn+1共cn−1+ cn+1兲 + W1共兩cn−1兩2cn−1− c¯n−1cn
2 − 2兩cn兩2cn−1+ 2兩cn兩2cn+1+ c¯n+1cn2−兩cn+1兩2cn+1兲, 共14兲
and in the case at hand W1⬇0.045.
Alternatively, this case can be realized by choosing even nonlinearity共= 1兲
G共x兲 = 0.0275 + 4.809 cos共2x兲 − 10 cos共4x兲. 共15兲
Now W1⬇0.012 and the lattice equation reads
ic˙n=0cn+1共cn−1+ cn+1兲 + W1共兩cn−1兩2cn−1+ c¯n−1cn2
+ 2兩cn兩2cn−1+ 2兩cn兩2cn+1+ c¯n+1cn2+兩cn+1兩2cn+1兲. 共16兲
Case 3. W0= W1= 0, W2⫽0 is obtained, for instance, for the even nonlinearity共= 1兲
G共x兲 = − 23.836 + 48.882 cos共2x兲 − 37.778 cos共4x兲,
共17兲 for which W2⬇−0.0136. The model 共10兲 is now simplified as ic˙n=0cn+1共cn−1+ cn+1兲 + W2共2兩cn−1兩2cn+ c¯ncn−1 2 + c¯ncn+1 2 + 2兩cn+1兩2cn兲. 共18兲
Case 4. W0= 0, W1,2⫽0 can be achieved by using an even nonlinearity共= 1兲
G共x兲 = 7.795 − 10 cos共2x兲. 共19兲
Now the overlap integrals are as follows: W1⬇0.0148 and
W2⬇0.0045. The lattice model then reads
ic˙n=0cn+1共cn−1+ cn+1兲 + W1共兩cn−1兩2cn−1+ c¯n−1cn2
+ 2兩cn兩2cn−1+ 2兩cn兩2cn+1+ c¯n+1cn2+兩cn+1兩2cn+1兲
Case 5.兩W1兩, 兩W2兩 Ⰶ兩W0兩 is the standard case of the on-site nonlinearity共the DNLS equation兲
ic˙n=0cn+1共cn−1+ cn+1兲 + W0兩cn兩2cn. 共21兲
This form of the lattice dynamical model is obtained for generic nonlinearities, Eq.共21兲 is well studied in the litera-ture共see, e.g., 关7兴兲 and that is why it will not be addressed in this paper.
III. MODULATIONAL INSTABILITY
As is customary we start with the analysis of the modu-lational instability of plane-wave solutions of Eq.共10兲 共for a general study of the modulational instability of the plane-wave background in the DNLS-type equations, see, e.g., 关28–30兴兲. This stability analysis is performed not only be-cause it is of interest in its own right but also bebe-cause the solitary wave solutions that we plan on constructing for the above presented lattices should be produced on a dynami-cally stable background. Using the plane-wave solution of the form
cn= Fei共qn−t兲, 共22兲
where F is a constant amplitude, we obtain the dispersion relation共in the absence of the previously considered cubic onsite terms兲
=0+ 21cos共q兲 + 4W1F2共+ 1兲cos共q兲 + 2W2F2关2
+ cos共2q兲兴. 共23兲
To examine the linear stability, we perturb the plane-wave solution in the form
cn=共F + Aei共Qn−⍀t兲+ B¯ e−i共Qn−⍀t兲兲ei共qn−t兲, 共24兲
with兩A兩, 兩B兩Ⰶ兩F兩 and linearize with respect to A and B. As a result we obtain two branches of the linear excitations ⍀1,2共Q兲, ⍀1,2= M−⫾
冑
共M+− D兲2− F4⌬, 共25兲 where D =−0− 4W1共1 +兲F2cos共q兲 − 4W2F2关cos共Q兲 + 1兴, M−= − 2关1+ 2W1共1 +兲F2兴sin共Q兲sin共q兲 − 4W2F2sin共Q兲sin共2q兲, M+= 2关1+ 2W1共1 +兲F2兴cos共Q兲cos共q兲 + 4W2F2cos共Q兲cos共2q兲, ⌬ = 4兵W1共1 +兲关1 + cos共Q兲兴cos共q兲 + W2关2 cos共Q兲 + cos共2q兲兴其2+ 4共1 −兲2sin2共q兲W 1 2关1 − cos共Q兲兴2. Let us consider different special cases for = −1 共the case= 1 is investigated in关22兴兲.
共a兲 Homogeneous background is described by q=0. Now the two branches of the solutions collapse and the dispersion relation acquires the form
⍀2= 16共 1+ 2W2F2兲sin2
冉
Q 2冊
冋
共1+ 6W2F 2兲sin2冉
Q 2冊
− 3W2F2册
. 共26兲The carrier field is stable if and only if the two following conditions are satisfied:
共1+ 2W2F2兲W2艋 0,
共1+ 2W2F2兲共1+ 3W2F2兲 艌 0, 共27兲 where the first of these conditions demands non-negativity of the coefficient k in the expansion of Eq.共26兲 of the form of ⍀2共Q兲=kQ2, which is valid for small Q. The second condi-tion demands non-negativity of⍀2共Q兲 at the zone boundary,
Q =.
We point out that the long wavelength excitations’ group velocity dispersion is given by
冏
2⍀Q2
冏
Q=0= 2W2关2W2F4+共8W2− 6 +1兲F2+ 41兴, 共28兲 and takes zero values for
F =
再
6 −1− 8W2 4W2 ⫾冑
冉
6 −1− 8W2 4W2冊
2 −21 W2冎
1/2 .This last condition determines the domain of the parameters where shock waves can be observed关31兴.
共b兲 Staggered background corresponds to q=. The dis-persion relation is ⍀2= 16共 1− 2W2F2兲sin2
冉
Q 2冊
冋
共1− 6W2F 2兲sin2冉
Q 2冊
+ 3W2F2册
. 共29兲Similar to Eq.共27兲 we introduce stability criteria in the form 共1− 2W2F2兲W2艌 0,
共1− 2W2F2兲共1− 3W2F2兲 艌 0. 共30兲 Note that the dispersion relations for q = 0, do not de-pend on the coefficient W1.
共c兲 Phase alternating background where q=/2. The dis-persion relation is ⍀1,2= − 21sin共Q兲 ⫾ 4
冑
2F2sin冉
Q 2冊
冑
共W2 2+ W 1 2兲cos共Q兲 − W 1 2. 共31兲 The condition for the presence of modulational instability in this case iscos共Q兲 ⬍ W1 2
W12+ W22
IV. QUASILINEAR MODEL (CASE 1)
Let us now turn to the quasilinear model of case 1. We notice that the term “quasilinear” is used in order to empha-size the existence of solely higher order contributions from more distant neighbors which are small but not necessarily zero. This provides us with an excellent benchmark of our derivation since in the discrete linear case, the dynamical equation can be solved explicitly and subsequently compared to the full results of the original partial differential equation 共for which the discrete model was developed as an approxi-mation兲. In particular, it is well known that the linear discrete case, with a compactum of initial data cn共0兲=A␦n0 has a
solution of the form
cn共t兲 = A共− i兲nexp共− i0t兲Jn共21t兲, 共33兲 where Jn is the Bessel function of order n.
We have tested this analytical prediction of the discrete
model in the partial differential equation共1兲 with the
“non-linear potential” of Eq. 共12兲. The results of our numerical simulations can be found in Fig.1, which highlights the ex-cellent agreement between the analytical and the numerical results. This also serves to showcase the accuracy of the reduction via the tight-binding approximation of the original partial differential equation. We will hereafter focus more on the details of these discrete models and of their solitary wave solutions.
The remarkable accuracy of the tight-binding model in the case at hand can be easily understood. Indeed, we are dealing with a perfect lattice 共i.e., having no defects兲. Inter-band transitions, which are the cause of the failure of the one-band approximation when they exist, are only due to the nonlinear coupling of bands, and exactly this factor is anomalously small for the chosen nonlinearity. The first indication on this fact is given by the zero contributions of W0,1,2. Next, due to the symmetry of the Wannier functions, and the symmetry of the nonlinearity, one concludes that W␣␣nnnn = 0 for all␣ and
, i.e., there exists no tunneling between the same sites of two different bands. Even more generally 共also due to the symmetry兲, W␣,,,␣+2␥n,n−m,n+m,n= 0 共for arbitrary integers , ␥, n,
and m兲. For completeness we have checked the numerical values of the other interband overlap integrals for the three lowest bands. The integrals greater than 0.01 are as follows 共the ones obtained by the symmetry reductions are not shown兲: W21110000= 0.0465, W22210000= −0.1360, W22111000= −0.0228,
W31111000= −0.0432, W33111000= −0.0301, W13331000= −0.0117. The coef-ficients describing energy transfer from the first to the second and the third bands are W21110000= 0.0465, W31110000= 0, W21111000 = 0.0077, W31111000= −0.0432, W21111100= 0.0004, and W31111100
= 0.0011, i.e., either have relatively small value or are iden-tically zero. This explains the high accuracy of the one-band approximation.
Finally, we notice that in the case at hand the dispersion relation共25兲 takes the form independent of the wave ampli-tude F,
⍀ = ⫾ 21关cos共q兲 − cos共Q ⫿ q兲兴, 共34兲 which is the dispersion relation for linear phonons, and hence no instabilities can occur.
V. NONLINEAR MODELS (CASES 2 AND 3)
The above confirmation of the high accuracy of the tight-binding model for the description of the pulse dynamics mo-tivated us to study the coherent structure solutions of Eq. 共10兲 and their properties for cases 2 and 3 singled out in Sec. II B. It is convenient to join the equations of cases 2 and 3 and to consider for= 1 the model
t x 0 20 40 60 80 100 −10 −5 0 5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t n 0 20 40 60 80 100 −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 |u 0 (t)| 2 t
FIG. 1. 共Color online兲 The first and second panels show the space-time contour plot of the solution in the PDE of Eq.共1兲 and
the discrete model analytical prediction 共33兲, respectively. In the
latter case, space is normalized over the period of the linear poten-tial, so that it can be compared with the lattice results. To accentuate the excellent agreement between analytical and numerical results, the third panel shows the time evolution of the amplitude at the central site of the configuration compared between the PDE numeri-cal result共solid line兲 and the discrete equation analytical result of Eq.共33兲 共dashed line兲. The two are practically indistinguishable.
ic˙n=0cn+1共cn−1+ cn+1兲 + W1共兩cn−1兩2cn−1+ c¯n−1cn 2 + 2兩cn兩2cn−1+ 2兩cn兩2cn+1+ c¯n+1cn 2 +兩cn+1兩2cn+1兲 + W2共2兩cn−1兩2cn+ c¯ncn−1 2 + c¯ncn+1 2 + 2兩c n+1兩2cn兲. 共35兲
Let us seek stationary solutions of Eq.共35兲 of the form
cn共t兲 = fne−it, 共36兲
with real fn. Using this ansatz, we obtain the equation for the
amplitudes 共0−兲fn+1共fn−1+ fn+1兲 + W1关fn−13 + fn+13 + 3fn 2共f n+1+ fn−1兲兴 + 3W2fn共fn−1 2 + fn+12 兲 = 0. 共37兲 We attempt to find the pulse 共bright soliton兲 and kink 共dark soliton兲 solutions. Our strategy in searching for these solutions will be as follows. We will first formulate the nec-essary conditions for the existence of the soliton solutions considering their carrier constant-amplitude field and also the soliton tail solutions. This will narrow the domain of param-eters where such solutions can be expected. Then, we will attempt to construct the desired soliton solutions and subse-quently study their stability.
The staggered and nonstaggered stationary solutions are connected by the following symmetry relation关25兴: if fnis a
solution of Eq. 共37兲 for definite W1, W2, and ⬍0 共⬎0兲, then 共−1兲nfn is a solution for W˜1= W1, W˜2= −W2, and˜ = 20−⬎0共˜⬍0兲. We also note that the stability analysis of any stationary solution can also be done, without loss of generality, for only, say, nonstaggered carrier field. This is so because the dynamics in the vicinity of the station-ary solution is governed by Eq.共51兲 共see below兲, which is invariant with respect to the following transformation: ⑀n
→共−1兲n⑀
n, fn→共−1兲nfn, →20−, W2→−W2, and t→ −t. Bearing this in mind, in the following we will discuss only stationary solutions with a nonstaggered background.
A. Constant-amplitude solution and soliton tails
A pulse solution must satisfy the boundary conditions
兩cn兩2→0 for n→ ⫾⬁, while for the kink solution one must
have兩cn兩2→F2⬎0 for n→ ⫾⬁. Thus, the existence and
sta-bility of the carrying field solution, described by formula 共22兲 with real F, is a necessary condition for the existence and stability of the soliton solutions.
One always has the trivial solution F = 0 and from the expression共23兲 one can have two nonzero solutions with
F2= −0− 21 8W1+ 6W2
, 共38兲
if the expression in the right-hand side of Eq.共38兲 is positive. In the case 8W1+ 6W2= 0, F can be arbitrary if=0+ 21, but we will not study this very special case.
Substituting Eq.共38兲 into Eq. 共25兲 we obtain the spectrum of the carrier field with F2⬎0, whose stability criteria 共in full analogy with Sec. III兲 are
关W1共−0+ 21兲 + W2共−0+1兲兴 ⫻共4W1+ 3W2兲共−0− 21兲 艋 0,
关W1共−0+ 21兲 + W2共−0+1兲兴
⫻关2W1共−0+ 21兲 + 3W2共−0兲兴 艌 0. 共39兲 From the asymptotic properties mentioned above, one can express the soliton tails as
fn⬃ F +n 共40兲
at兩n兩→⬁, where smalln are real and are independent on t.
Substituting Eq.共40兲 into Eq. 共37兲 and linearizing with re-spect tonone obtains
␥n−1+n+␥n+1= 0, 共41兲
with
=0−+ 12W1F2+ 6W2F2,
␥=1+ 6W1F2+ 6W2F2. 共42兲
Seeking solutions to Eq.共41兲 in the form
n⬃ C⫾zn, n→ ⫾ ⬁,
we come to the characteristic equation
␥z2+z +␥= 0. 共43兲
Thus, z is one of the roots z1,2,
z1= 1 z2 = −  2␥+
冑
2 4␥2− 1, 共44兲 providing兩n兩→0 as 兩n兩→⬁.For the soliton, which is either a homoclinic or a hetero-clinic of the map, generated by Eq.共37兲, n⬅0 must be a
hyperbolic point which happens only if the roots z1,2are real, i.e., if
2− 4␥2⬎ 0. 共45兲
The sign of z specifies the type of tail solution: z⬎0 corresponds to the tail decaying monotonically with distance from the soliton’s center, while z⬍0 means that the decaying tail solution oscillates near the carrier solution F.
The absolute value of z characterizes the degree of local-ization of the tail. If 兩z兩 is small, then the tail solution is weakly localized, otherwise it is strongly localized. We found that, in many cases, there is a correlation between the degree of localization of the tail solution and that of the soliton itself.
The tail solution was found from the linearized equation and it can only provide necessary conditions for the exis-tence of a soliton solution. The nonlinear terms, depending on their structure and the values of the corresponding coef-ficients, can either lead to unbounded solutions or to nonlo-calized solutions, or to the desired bounded and lononlo-calized soliton solutions.
Having a tail solution one can attempt to construct the corresponding soliton solution. To do so, we use the tail so-lution defined by Eq.共40兲 to set the initial values for fn−1and
fn and then find fn+1 from Eq. 共37兲, solving the cubic 共for
W2= 0兲 or the quadratic 共for W1= 0兲 algebraic equation. The proper choice of the integration constants C⫾systematically allows one to obtain equilibrium on-site or intersite soliton
solutions, if they exist关38兴. To conclude this section we sum-marize the necessary conditions for the existence of pulses and kinks.
Pulse in cases 2 and 3. Since the carrying field for pulses
with F = 0 always exists, there remains only one necessary condition, namely, the condition of the existence of the tail solution of Eq.共45兲. For F=0 this condition is satisfied for any W1and W2and for both cases 2 and 3 when
⬎0− 21= − 0.737 327 41⬅a,
⬍0+ 21= − 0.940 417 21⬅b. 共46兲
These conditions simply state that the localized pulses must be located outside the phonon band of the spectrum given by the interval关b,a兴.
The necessary conditions of the existence of a kink in-clude the condition of the existence of the carrier field with
F2⬎0, Eq. 共38兲, the stability condition for the carrier field, Eq.共39兲, and condition 共45兲 of the existence of the tail solu-tion.
Kink in case 2. All three necessary conditions are satisfied
when W1⬎0 and b⬍⬍a, and they are not satisfied
simultaneously for W1⬍0.
Kink in case 3. All three necessary conditions are satisfied
when W2⬎0 andb⬍⬍0, while for W2⬍0 they are sat-isfied for0⬍⬍a.
We also specify the frequencies at which z changes sign. In case 2 the frequencies are
0−共2/3兲1= − 0.805 024 01⬅c,
0+共2/3兲1= − 0.872 720 61⬅d, 共47兲
while in case 3 they are
0−1= − 0.788 099 86⬅e,
0+1= − 0.889 644 76⬅f. 共48兲
B. Soliton solutions
It is well known关25,33兴 that the standard DNLS equation 共21兲 possesses multiple branches of localized solutions which are parametrized by the frequency detuning outwards from the phonon band. Multiple branches of the stationary localized solutions were also obtained in a different model in the presence of both linear and nonlinear lattices关19兴. This allows one to conjecture that any of the lattices introduced in Sec. II should also possess multiple branches of the localized solutions and kinks共discrete dark solitons兲. Although a thor-ough study of each of the cases is by itself a nontrivial prob-lem that requires lengthy considerations, to present a pan-oramic view of the possible nonlinear modes in the above lattices, in the present paper we restrict our study to some representative examples, referring, in most cases, to the lowest branches.
Case 2共W2= 0兲. For the nonlinearity of case 2, using the method described in Sec. V A, we obtain examples of local-ized solutions of Eq. 共35兲, namely, pulse, staggered pulse,
kink, staggered kink, wave共an antidark structure in the form of an oscillatory wave on a nonzero background兲, and stag-gered wave. The nonstagstag-gered solutions are presented in Fig. 2 for the parameters: 共a兲 W1= −0.0112, = −0.99, 共b兲 W1 = 0.0112, = −0.9, 共c兲 W1= 0.0112, = −0.75. The corre-sponding staggered solutions can be constructed using the staggering transformation, described above. On-site equilib-rium structures are shown but one can also obtain the intersite ones.
In Fig.3, the regions of existence of various solutions are indicated on the parameter plane 共W1,兲. Solutions pre-sented in Figs. 2共a兲–2共c兲 have parameters shown by dots marked by the capital letters A – C, respectively. Recall that pulses can exist in the two frequency ranges⬍b and
⬎a for any W1. However, they were found only for W1 ⬍0, while for W1⬎0 the iterations initiated by the tail solu-tion resulted in unbounded structures. In the porsolu-tion of the plane with W1⬎0 and b⬍⬍c kinks 关see Fig. 2共b兲兴,
staggered waves were obtained. On the other hand, in the portion with W1⬎0 and d⬍⬍a, we could construct
staggered kinks and waves 关see panel 共c兲 of Fig. 2兴. It is important to note that兩z兩 is close to 1 near the lines=a
and=b, where pulses, kinks, and waves were found to be
wide; z diverges共or vanishes兲 at=cfor staggered carrier
field and at=dfor nonstaggered carrier field; in the range
ofd⬍⬍c, z is always negative and close to 0, resulting
in rapidly oscillating or sharply localized共and typically un-stable兲 solutions found from the tail construction.
Case 3共W1= 0兲. In case 3, three examples of the nonstag-gered stationary soliton solutions presented in Fig. 4 were FIG. 2. Equilibrium soliton solutions found in case 2共W2= 0兲: 共a兲 pulse, 共b兲 kink, and 共c兲 antidark structure with oscillatory wave structure around the background, hereafter termed wave. On-site structures are presented here, but intersite ones can also be con-structed. Solutions in共a兲 and 共b兲 can be stable but we were unable to find a stable structure of the form presented in共c兲 共see Sec. V C兲. Model parameters corresponding to panels共a兲–共c兲 are depicted by dots in Fig.3marked by the capital letters A – C, respectively. Pa-rameters are共a兲 W1= −0.0112,=−0.99, 共b兲 W1= 0.0112,=−0.9, and共c兲 W1= 0.0112,=−0.75.
found. Shown are 共a兲 pulse, 共b兲 kink, and 共c兲 kink with os-cillatory tail, referred to as osos-cillatory kink. On-site struc-tures are presented here, but intersite ones can also be con-structed. Model parameters corresponding to panels 共a兲–共c兲
of Fig.4are depicted by dots in Fig.5marked by the capital letters A – C, respectively. Parameters for the nonstaggered solutions are共a兲 W2= −0.0136,= −0.9704,共b兲 W2= 0.0136,
= −0.9, and 共c兲 W2= 0.0136, = −0.85, while the corre-sponding staggered solutions, as in case 2, can be obtained using the staggering transformation.
The pulse tail solution共with z⬎0兲 exists for ⬍b but
the pulse itself exists in this region only for W2⬍0, while for positive W2we obtained unbounded solutions. Similarly, the staggered pulse tail solution共with z⬍0兲 exists for⬎abut
the staggered pulse itself exists in this region only for W2 ⬎0, while negative W2leads to unbounded solutions.
The kink exists for W2⬎0 and b⬍⬍f, while the
staggered kink exists for W2⬍0 ande⬍⬍a. Finally, the
oscillatory kink exists for W2⬎0 andf⬍⬍0, while the oscillatory staggered kink exists for W2⬍0 and 0⬍ ⬍e.
In Figs. 6 and 7 we contrast the behavior of pulses in cases 2 and 3, respectively. In both figures 共a兲 shows the space-time evolution of 兩cn共t兲兩2, while 共b兲 shows the time
variation of the norm of the central particles. On purpose, we did not optimize the choice of C⫾to get equilibrium on-site or intersite solutions. As a result, the ensuing profiles are nonstationary and, due to the presence of the Peierls-Nabarro potential, they oscillate in the vicinity of stable configura-tions. It turns out that in case 2共case 3兲 the intersite 共on-site兲 configuration is stable. This conclusion will be confirmed in Sec. V C. This illustrates the interesting phenomenon of po-tential inversion of stability共cf. 关22兴兲 in comparison with the standard DNLS mode 关7兴. Parameters in Fig. 6 are W1 = −0.012, W2= 0, and = −0.99, which corresponds to the point A in Fig. 3. Parameters in Fig. 7 are W1= 0, W2 FIG. 3. Parameter plane共W1,兲 in case 2 共W2= 0兲 with
indi-cated regions of existence of three types of solitons shown in Fig.2
and their staggered analogs. The pulse solution exists for W1⬍0 and ⬍b, while the staggered pulse exists for W1⬍0 and ⬎a. The kink and staggered wave can exist for W1⬎0 andb
⬍⬍c, while the staggered kink and the wave can exist for W1 ⬎0 andd⬍⬍a. The values of special frequenciesatodare
given by Eqs.共46兲 and 共47兲.
FIG. 4. Equilibrium soliton solutions obtained in case 3 with W1= 0:共a兲 pulse, 共b兲 kink, and 共c兲 kink with oscillatory tail, called oscillatory kink. On-site structures are presented here, but intersite ones can also be constructed. All these solutions can be stable, as will be shown in Sec. V C. Model parameters corresponding to panels共a兲–共c兲 are depicted by dots in Fig.5marked by the capital Latin letters A – C, respectively. Parameters are 共a兲 W2= −0.0136, =−0.9704, 共b兲 W2= 0.0136, =−0.9, and 共c兲 W2= 0.0136, = −0.85.
FIG. 5. Parameter plane共W2,兲 in case 3 共W1= 0兲 with indi-cated regions of existence of three types of solitons shown in Fig.4
and their staggered analog. The pulse exists for W2⬍0 and ⬍b, while the staggered pulse exists for W2⬎0 and⬎a. The
kink exists for W2⬎0 andb⬍⬍f, while the staggered kink
exists for W2⬍0 ande⬍⬍a. Finally, the oscillatory kink ex-ists for W2⬎0 andf⬍⬍0, while the oscillatory staggered kink exists for W2⬍0 and0⬍⬍e. Values of special frequencies are given by Eqs.共46兲 and 共48兲.
= −0.0136, and= −0.9704, which corresponds to the point
A in Fig.5.
Similar results for dark solitons are presented in Fig. 8 共kink in case 2兲 and Fig.9 共kink with oscillatory tail in case 3兲. One can see that in both cases the on-site structures are stable and this will be confirmed in Sec. V C. Parameters in Fig. 8 are W1= 0.012, W2= 0, and = −0.9, which corre-sponds to the point C in Fig. 3. Parameters in Fig. 9 are
W1= 0, W2= 0.0136, and= −0.88.
C. Stability of soliton solutions
Let us now study the stability of the stationary soliton solutions of the form of Eq.共36兲 described in Sec. V B. We consider the following perturbed form of the solutions:
cn共t兲 = 关fn+⑀n共t兲兴e−it, 共49兲
where the small complex perturbation⑀n共t兲 is expressed as
follows: FIG. 6. Pulse in case 2 共W2= 0兲. 共a兲 Space-time evolution of
兩cn共t兲兩2showing the dynamics of the pulse placed at t = 0
asymmetri-cally with respect to the lattice. The pulse undergoes periodic oscil-lations in the vicinity of the stable intersite configuration.共b兲 Time variation of the norm of the four central particles. Parameters: W1 = −0.012, W2= 0, and=−0.99; these correspond to the point A in Fig.3.
FIG. 7. Pulse in case 3 共W1= 0兲. 共a兲 Space-time evolution of 兩cn共t兲兩2showing the dynamics of the pulse placed at t = 0
asymmetri-cally with respect to the lattice. The pulse undergoes periodic oscil-lations in the vicinity of the stable on-site configuration.共b兲 Time variation of the norm of the central particle and its two nearest neighbors. Parameters: W1= 0, W2= −0.0136, and=−0.9704; these correspond to the point A in Fig.5.
FIG. 8. Kink in case 2 共W2= 0兲. 共a兲 Space-time evolution of 兩cn共t兲兩2showing the dynamics of the kink placed at t = 0
asymmetri-cally with respect to the lattice. The kink undergoes periodic oscil-lations in the vicinity of the stable on-site configuration.共b兲 Time variation of the norm of the central particle and its two nearest neighbors. Parameters: W1= 0.012, W2= 0, and=−0.9, which cor-responds to the point B in Fig.3.
FIG. 9. Kink with oscillatory tail in case 3共W1= 0兲. 共a兲 Space-time evolution of 兩cn共t兲兩2 showing the dynamics of the kink with
oscillatory tail placed at t = 0 asymmetrically with respect to the lattice. The kink with oscillatory tail undergoes periodic oscillations in the vicinity of the stable on-site configuration.共b兲 Time variation of the norm of the central particle and its two nearest neighbors. Parameters: W1= 0, W2= 0.0136, =−0.88. The kink with oscilla-tory tail has frequency close to f= −0.8896 and it becomes un-stable far from this line共see Fig.15兲.
⑀n共t兲 = an共t兲 + ibn共t兲. 共50兲
Substituting Eq.共49兲 into Eq. 共10兲 we find that⑀n共t兲 is
gov-erned by the following linearized equation:
i⑀˙n=共0−兲⑀n+1共⑀n−1+⑀n+1兲 + W1关fn−12 Zn−1 + 2fn共fn−1+ fn+1兲Zn+ fn+1 2 Zn+1+ fn 2共Z n−1+ Zn+1兲兴 + W2关2fnfn−1Zn−1+共fn−1 2 + fn+12 兲Zn+ 2fnfn+1Zn+1兴, 共51兲 where Zn= 3an+ ibn. Then, separating real and imaginary
parts of Eq.共51兲 we derive the following system:
冉
b˙ a˙冊
=冉
0 K J 0冊冉
b a冊
, 共52兲where vectors a and b contain anand bn, respectively, while
the nonzero coefficients of matrices K and J are given by
Kn,n−1= −1− 3W1共fn−1 2 + fn 2兲 − 6W 2fn−1fn, Kn,n=−0− 6W1fn共fn−1+ fn+1兲 − 3W2共fn−12 + fn+12 兲, Kn,n+1= −1− 3W1共fn2+ fn+12 兲 − 6W2fnfn+1, Jn,n−1=1+ W1共fn−1 2 + fn 2兲 + 2W 2fn−1fn, Jn,n=0−+ 2W1fn共fn−1+ fn+1兲 + W2共fn−1 2 + fn+12 兲, Jn,n+1=1+ W1共fn2+ fn+12 兲 + 2W2fnfn+1. 共53兲
In the above expressions, n = 1 , . . . ,N, where N is the num-ber of lattice points. For pulses and kinks, we used periodic and antiperiodic boundary conditions, respectively.
A stationary solution is characterized as linearly stable if and only if the eigenvalue problem
冉
0 K J 0冊冉
b a冊
=␥冉
b a冊
共54兲results in nonpositive real parts of all eigenvalues␥. The results of the stability analysis for the equilibrium structures reported in Sec. V B are presented in Figs.10–12 for case 2 and in Figs.13–15for case 3. The presented spec-tra contain共i兲 the vibration frequencies of the homogeneous
background given by Eq. 共26兲; 共ii兲 a pair of zero-frequency modes corresponding to the invariance with respect to the phase shift;共iii兲 they also can include soliton internal modes falling outside the phonon band, see, e.g., Fig.14共a兲. As was already mentioned, the spectra of stable structures do not possess eigenvalues with positive real parts, while those of the unstable ones have such eigenvalues. Now we turn to the discussion and comparison of the spectra in cases 2 and 3.
Case 2共W2= 0兲. Spectra of the on-site and intersite pulses are presented in Figs.10共a兲and10共b兲, respectively. Interest-ingly, the intersite configuration is stable while the on-site one is unstable. This type of instability is typical for the discrete system with Peierls-Nabarro potential, although in the standard cubic onsite nonlinearity case, the results are entirely reversed in comparison to the present case关7兴 共e.g., the on-site pulse is stable, while the intersite features a real eigenvalue pair兲. This indicates that case 2 nonlinearity re-sults in the shape of the Peierls-Nabarro potential having a maximum共minimum兲 for the on-site 共intersite兲 pulses. No-tice that as discussed in关22兴, such inversions of stability may occur in such generalized models, upon varying their rel-evant parameters共such as W1 in the present case兲. We will see that for case 3 nonlinearity the situation for the pulse is exactly the opposite. The parameters used in this case are
W1= −0.012, W2= 0,= −0.97, corresponding to the point A in Fig.3.
Figure11shows results for the kink structures: the on-site kink in共a兲 is stable while the intersite one in 共b兲 is unstable. Here the location of maxima and minima of the Peierls-Nabarro potential is the same as in the classical discretiza-FIG. 10. Spectra of共a兲 unstable on-site and 共b兲 stable intersite
pulses. The parameters 共case 2兲: W1= −0.012, W2= 0, and = −0.99 correspond to the point A in Fig.3.
FIG. 11. Spectra of共a兲 stable on-site and 共b兲 unstable intersite kinks. The parameters 共case 2兲: W1= 0.012, W2= 0, and =−0.9 correspond to the point B in Fig.3.
FIG. 12. Spectra of共a兲 unstable on-site and 共b兲 unstable intersite wave. The parameters 共case 2兲: W1= 0.012, W2= 0, and=−0.75 correspond to the point C in Fig.3.
tion. The parameters W1= 0.012, W2= 0.0, and= −0.9 cor-respond to the point B in Fig.3.
Finally, in Fig.12we show that共a兲 the on-site wave and 共b兲 the intersite wave are both unstable. The parameters W1 = 0.012, W2= 0, and = −0.75 correspond to the point C in Fig.3.
Case 3共W1= 0兲. Results of the stability analysis are pre-sented in Figs. 13–15 for the three soliton solutions dis-played in panels共a兲–共c兲 of Fig.4, respectively. The left pan-els show the spectra of the on-site structures and the right panels show the same for the corresponding intersite struc-tures. One can see that, in contrast to case 2, where the intersite pulse was found to be stable, in case 3 the intersite structures are always unstable共this is analogous to the case of the standard cubic discrete model with the on-site nonlin-earity兲. This indicates that in case 3, the on-site 共intersite兲 structures are situated in the wells 共on the peaks兲 of the Peierls-Nabarro potential. On the other hand, panels 共a兲 in Figs.13–15present purely imaginary spectra for the on-site configurations, and this indicates that, for the corresponding values of model parameters, all three types of equilibrium solutions are stable. However, in Fig.15we demonstrate that the on-site configuration of the oscillatory kink is stable at
= 0.88 关see panel 共a兲兴 but it can become unstable, e.g., at
= 0.85 关see panel 共c兲兴 with other parameters being un-changed and equal to W1= 0, W2= 0.0136.
VI. CONCLUSIONS
In this work, we have illustrated the potential for formu-lation of a rich variety of tight-binding nonlinear lattice
dy-namical models, stemming from the complex interplay of linear and nonlinear periodic potentials in the nonlinear Schrödinger equation. We have examined various particular possibilities, including quasilinear models, as well as strongly nonlinear models where the nature of the coupling between the neighbors is itself nonlinear. Furthermore, in the nonlinear ones among our models, we have studied the po-tential for the existence and stability, as well as the dynamics of localized solutions. More specifically, we have reported that the discrete model of Eq.共35兲 with W1⫽0, W2= 0共case 2兲, and W1= 0 and W2⫽0 共case 3兲 supports a number of localized stationary solutions depicted in Figs.2 and 4. In-terestingly, the nonlinearity of case 2 results in stable
inter-site pulse and stable on-inter-site kink solutions共both in staggered
and nonstaggered forms兲. On the other hand, the nonlinearity of the case 3 supports only on-site stable localized solutions of three different types, namely, pulses, kinks, and kinks with oscillatory tails 共all in both staggered and nonstaggered forms兲. These results suggest that in the discrete models with nonlinear terms including interactions between nearest neighbors the profile of the Peierls-Nabarro potential can change qualitatively depending on the structure of the non-linear terms and on the type of the coherent structure. In fact, it has been demonstrated that there exists a wide class of nonintegrable discrete models of this sort where the Peierls-Nabarro potential is precisely equal to zero and equilibrium stationary solutions can be placed anywhere with respect to the lattice points关34兴.
At the same time, it should be mentioned that we have left open an important question about the mathematical definition of the domains of the applicability of the suggested lattice models, and more specifically of the one-band 共tight-FIG. 13. Spectra of共a兲 stable on-site and 共b兲 unstable intersite
pulses. Oscillatory motion of the pulse in the vicinity of the on-site configuration is shown in Fig.6. The parameters 共case 3兲 W1= 0, W2= −0.0136,=−0.9704 correspond to the point A in Fig.5.
FIG. 14. Spectra of共a兲 stable on-site and 共b兲 unstable intersite kinks. The parameter values 共case 3兲 W1= 0, W2= 0.0136, and=−0.9 correspond to the point B in Fig.5.
FIG. 15. The top panels show spectra of共a兲 stable on-site and 共b兲 unstable intersite oscillatory kinks with=−0.88. The bottom panels show the same for=−0.85 when both on-site and intersite oscillatory kinks become unstable. Note that the type of instability in 共c兲 is different from that in 共d兲. In 共c兲 there is a quartet of eigenvalues, while in共d兲 there is a pair of purely real eigenvalues. The rest of the parameters are共case 3兲 W1= 0, W2= 0.0136.
binding兲 approximation. This is a highly nontrivial question, that has not been fully addressed so far even for the more standard mapping of the NLS equation to the DNLS equation 关20,35,36兴. This question, however, goes beyond the scope of the present paper and is an important open problem. It is relevant to emphasize, in that connection, the following facts:
共1兲 The mapping of Eq. 共1兲 to Eq. 共5兲 is exact, as it is based on the expansion over the complete set of the Wannier functions. This is also the reason of the generality of the method, which can be applied to other physical systems共as a matter of fact it is implicitly applied when one uses the tight-binding approximation for the description of the light propa-gation in arrays of waveguides兲. It is particularly well suited to problems with linear共as well as nonlinear兲 operators with periodic coefficients.
共2兲 As it has been established in 关20兴, the one-band ap-proximation has its domain of validity, although, to the best of the authors’ knowledge, it has not been established rigor-ously and under general conditions so far.
共3兲 Even in the case when the tight-binding approximation 共10兲 does not provide the desired quantitative accuracy, it still allows for the qualitative prediction of several phenom-ena, including the existence, but also importantly the general
stability properties of the localized modes of the system. Another natural extension of the present work would be to consider similar types of reductions in higher dimensional settings and to examine in detail the particular localized so-lutions that may emerge in the resulting lattice models. In particular, higher dimensionality may offer the potential for solutions with topological charge; it would therefore be rel-evant to examine under what conditions such solutions may be stable and how the relevant results relate to the original continuum model.
ACKNOWLEDGMENTS
Y.V.B. was supported by the FCT Grant No. SFRH/PD/ 20292/2004. The work of P.G.K. is supported by Grants No. NSF-DMS-0505663, and No. NSF-DMS-0619492, and NSF-CAREER and by the University of Massachusetts. V.V.K. acknowledges support from Ministerio de Educación y Ciencia 共MEC, Spain兲 under Grant No. SAB2005-0195. The work of Y.V.B. and V.V.K. was supported by the FCT and European program FEDER under Grant No. POCI/FIS/ 56237/2004. S.V.D. gratefully acknowledges the financial support provided by the Russian Foundation for Basic Research, Grant No. 07-08-12152.
关1兴 O. Madelung, Introduction to Solid State Theory 共Springer-Verlag, Berlin, 1978兲.
关2兴 D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 共London兲 424, 817 共2003兲; J. W. Fleischer et al., Opt. Express
13, 1780共2005兲.
关3兴 D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 共1988兲.
关4兴 H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383共1998兲.
关5兴 V. A. Brazhnyi and V. V. Konotop, Mod. Phys. Lett. B 18, 627 共2004兲.
关6兴 O. Morch and M. Oberthaler, Rev. Mod. Phys. 78, 179 共2006兲. 关7兴 P. G. Kevrekidis, K. Ø. Rasmussen, and A. R. Bishop, Int. J.
Mod. Phys. B 15, 2833共2001兲.
关8兴 A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31共2003兲. 关9兴 R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg,
and Y. Silberberg, Phys. Rev. Lett. 83, 2726共1999兲.
关10兴 J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchi-son, Phys. Rev. Lett. 92, 163902共2004兲.
关11兴 F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606共2001兲; A. Smerzi, A. Trombettoni, P. G. Kevrekidis, and A. R. Bishop, Phys. Rev. Lett. 89, 170402共2002兲.
关12兴 F. S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, and M. Inguscio, New J. Phys. 5, 71共2003兲.
关13兴 Joachim Meier, Jared Hudock, Demetrios Christodoulides, George Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907共2003兲; R. L Horne, P. G. Kevrekidis, N. Whitaker, Phys. Rev. E 73, 066601共2006兲.
关14兴 S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stege-man, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901共2006兲; Xiaosheng Wang, Anna Bezryadina, Zhigang Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, ibid. 98, 123903共2007兲. 关15兴 D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Yu. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, Phys. Rev. Lett. 92, 123903共2004兲; J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, ibid. 92, 123904共2004兲.
关16兴 D. E. Pelinovsky, L. Brzozowski, and E. H. Sargent, Phys. Rev. E 62, R4536共2000兲.
关17兴 G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 1 共2006兲; Y. Sivan, G. Fibich, and M. I. Weinstein, Phys. Rev. Lett. 97, 193902共2006兲.
关18兴 F. Abdullaev and J. Garnier, Phys. Rev. A 72, 061605共R兲 共2005兲; G. Theocharis, P. Schmelcher, P. G. Kevrekidis, and D. J. Frantzeskakis, ibid. 72, 033614共2005兲; J. Garnier and F. K. Abdullaev, ibid. 74, 013604共2006兲; H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 共2005兲; 73, 026601 共2006兲; J. Belmonte-Beitia, V. M. Pérez-García, V. Veksler-chik, and P. J. Torres, Phys. Rev. Lett. 98, 064102共2007兲; F. Kh. Abdullaev, A. A. Abdumalikov, and R. M. Galimzyanov, Phys. Lett. A 367, 149共2007兲.
关19兴 Yu. V. Bludov and V. V. Konotop, Phys. Rev. A 74, 043616 共2006兲.
关20兴 G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Sal-erno, Phys. Rev. E 66, 046608共2002兲.
关21兴 V. V. Konotop and S. Takeno, Phys. Rev. B 55, 11342 共1997兲; Physica D 113, 261共1998兲.
056606共2003兲.
关23兴 M. Öster, Yu. B. Gaididei, M. Johansson, and P. L. Chris-tiansen, Physica D 198, 29共2004兲.
关24兴 A. Smerzi and A. Trombettoni, Phys. Rev. A 68, 023613 共2003兲; C. Menotti, A. Smerzi, and A. Trombettoni, New J. Phys. 5, 112共2003兲.
关25兴 G. L. Alfimov, V. A. Brazhnyi, and V. V. Konotop, Physica D
194, 127共2004兲.
关26兴 D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Physica D 212, 1共2005兲.
关27兴 W. Kohn, Phys. Rev. 115, 809 共1959兲.
关28兴 Yu. S. Kivshar and M. Salerno, Phys. Rev. E 49, 3543 共1994兲. 关29兴 V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 共1999兲. 关30兴 F. Kh. Abdullaev, S. A. Darmanyan, and J. Garnier, Prog. Opt.
44, 303共2002兲.
关31兴 V. V. Konotop and M. Salerno, Phys. Rev. E 55, 4706 共1997兲;
56, 3611共1997兲; V. V. Konotop, Chaos, Solitons Fractals 11,
153共2000兲.
关32兴 G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett.
58, 7共2002兲.
关33兴 E. W. Laedke, O. Kluth, and K. H. Spatschek, Phys. Rev. E
54, 4299共1996兲.
关34兴 P. G. Kevrekidis, S. V. Dmitriev, and A. A. Sukhorukov, Math. Comput. Simul. 74, 343共2007兲; S. V. Dmitriev, P. G. Kevreki-dis, N. Yoshikawa, and D. J. Frantzeskakis, J. Phys. A 40, 1727共2007兲; S. V. Dmitriev, P. G. Kevrekidis, A. A. Sukho-rukov, N. Yoshikawa, and S. Takeno, Phys. Lett. A 356, 324 共2006兲; 共see also e-print arXiv:nlin.PS/0603047, with corrected misprints兲; D. E. Pelinovsky, Nonlinearity 19, 2695 共2006兲; A. Khare, S. V. Dmitriev, and A. Saxena, J. Phys. A: Math. Theor.
40, 11301共2007兲.
关35兴 M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 共2003兲.
关36兴 M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems共Cambridge Uni-versity Press, Cambridge, England, 2004兲.
关37兴 The numerical values of the Fourier coefficients, of the Wan-nier functions, and of the overlap integrals presented in this paper were obtained by the software developed by G. L. Alfi-mov.
关38兴 This is a variant of the shooting method used for obtaining localized modes in continuous models关5,19,25,32兴.
