the appearance of gap solitons in a nonlinear schrödinger lattice
TRANSCRIPT
The appearance of gap solitons in a nonlinear
Schrodinger lattice
L. Kroon a,b, M. Johansson a,∗, A.S. Kovalev c , E.Yu. Malyuta a,d
aDepartment of Physics, Chemistry and Biology, Linkoping University, SE-581 83 Linkoping, SwedenbSwedish Defence Research Agency (FOI), P.O. Box 1165, SE-581 11 Linkoping, Sweden
cB. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences ofUkraine, pr. Lenina 47, Kharkov 61103, Ukraine
dElectrophysical Scientific and Technical Center of the National Academy of Sciences of Ukraine, ul.Chaιkovskogo 28, Kharkov 61002, Ukraine
Abstract
We study the appearance of discrete gap solitons in a nonlinear Schrodinger model with a peri-odic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit.For finite lattices supporting an anti-phase (q = π/2) gap edge phonon as an anharmonic stand-ing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchforkbifurcations from the gap edge. Analytically, modulational instabilities between pair of bifurca-tion points on this “nonlinear gap boundary” are found in terms of critical gap widths, turningto zero in the infinite size limit, which are associated with the birth of the localized soliton aswell as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in excitingsoliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensatesin periodic potentials. For lattices whose gap edge phonon only asymptotically approaches theanti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to thebirth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. Theinstability-induced dynamics of the localized soliton in the gap regime is found to thermalizeaccording to the Gibbsian equilibrium distribution, while the spontaneous formation of persist-ing intrinsically localized modes (discrete breathers) from the extended out-gap soliton revealsa phase transition of the solution.
Key words: Discrete gap solitons, bifurcations, linear stability, thermalizationPACS: 42.65.Wi, 42.82.Et, 63.20.Pw, 63.20.Ry
∗ Corresponding author.Email addresses: [email protected] (L. Kroon), [email protected] (M. Johansson),
[email protected] (A.S. Kovalev).
Preprint submitted to Physica D 29 August 2009
1. Introduction
The phenomenon of self-localization in lattice dynamical systems [1,2] is subject to agrowing interest in the physical society. For example, the emergence of coherent struc-tures in coupled waveguide arrays and photonic crystals in nonlinear optics [3–5] andBose-Einstein condensates in optical lattices [6] are important applications. Balancingnonlinearity and dispersion can give rise to localized time-periodic (breathing) excita-tions that are dynamically stable and thereby being able to trap conserved quantitiesover long periods of time [7]. Apart from the delicate question of mobility, the formal ex-istence of exact discrete breathers (see, e.g., Refs. [8–10] for reviews) has been establishedfor a wide class of anharmonic Hamiltonian lattices [11].
To ensure the survival of a spatially localized solution any harmonic generation of itsfrequency must be non-resonant with the linearized (continuous) spectrum. Endowingthe lattice with a periodicity gives rise to a spectral gap, separating two branches ofthe linear dispersion relation, in which discrete gap breathers can exist. A simple modelfor studying their existence is the discrete nonlinear Schrodinger (DNLS) equation withan alternating on-site potential [12]. Besides being a nonintegrable discretization of thenonlinear Schrodinger (NLS) equation it arises as the envelope wave reduction of generalnonlinear Klein-Gordon lattices [13,14], describing their small-amplitude dynamics, butwhere the periodic orbits can be made time-independent by transforming into a frameof reference rotating with a single frequency (see Refs. [15,16] for reviews).
It is the purpose of this study to shed light on the very appearance of anharmonic gapmodes in periodic nonlinear lattices, that in some approximation are described by thestationary solutions to the DNLS equation, of which the energetically favourable (not theground state though) state is referred to as the (stationary) discrete gap soliton. In anoptical setting this accounts for the concept of the gap (Bragg) soliton, first predicted inRef. [17], studied in modulated arrays of weakly coupled waveguides [18], which indeedhas been observed experimentally [19]. The experimental observation of Bose-Einsteingap solitons [20] in periodic potentials can also be mentioned in this context.
In continuous models gap solitons delocalize and vanish at one gap boundary, buttransform into out-gap solitons with frequencies in the linear spectrum at the other [21].To give a picture of their behaviour in a discrete model, which is described in Section 2,an analysis in vein of the pioneering work [2] is presented in Section 3 for finite lattices ofincreasing sizes, aiming at the limit of an infinite system. Besides the fundamental sideof the issue, the results for small systems are intrinsically interesting when consideredas vibrational modes in ring-like molecules. As mentioned in [22], magnetic moleculeswith alternating spins, like MnR6 compounds [23], are particular examples of finite-size discrete modulated systems for which the dynamics to some approximation can bedescribed within a DNLS model. A preliminary version of some of the results for the4-site system (Section 3.1) was reported in [24] (see also a related work [22]). However,it is one of the fundamental aims of the present work to connect the results of [22,24],discussing the smallest possible system where a solution with gap-soliton characteristicscan be identified, to the earlier results of [12], discussing very large (ideally infinite)lattices. Thus, in Section 3 we will combine analytical and numerical results to give acomplete description of the bifurcation scenarios leading to the emergence of gap modes
for any ring-like system with an even number of lattice sites (to avoid effects of boundary
2
mismatch, we do not here consider odd-sized lattices). As we will see, there are importantdistinctions between lattices of size M = 4m (Section 3.2) and M = 4m+2 (Section 3.3),and furthermore the 6-site system (Section 3.4) also turns out to show some distinctivefeatures.
Another important issue for the role of localized excitations in real systems is gener-ally, whether they may appear dynamically, e.g., whether they form spontaneously frominstabilities of extended solutions, and whether they may survive in a thermal equilib-rium state (see, e.g., the review [10] for discussions on these issues). Thus, a second aimof this paper is to illustrate the role of the gap solitons in this context. In Section 4,the dynamics resulting from (modulational) instabilities is discussed in connection withstatistical mechanics in the thermodynamic limit. We here extend earlier results fromRefs. [25–28] for the nonmodulated (gapless) DNLS model, showing the existence of aphase transition between a Gibbsian equilibrium state and another phase of statisti-cal localization with spontaneous formation of persistent localized modes, to the moregeneral case with on-site modulation studied in the present work. Here, one of our mainnovel results is, that although localized gap solitons may appear spontaneously as reason-ably long-lived objects in the instability-induced dynamics of unstable gap-edge modes,they are only metastable and typically thermalize into the Gibbsian equilibrium in thelong-time dynamics.
Finally, Section 5 summarizes our findings.
2. Model
We consider a closed chain (ring) of M coupled anharmonic oscillators (numbered asn = 1, 2, 3, . . . ,M) with a periodic potential denoted by Vn. The on-site modulation ischosen such that Vn = V1 (Vn = V2) when the site index n is an odd (even) number,where V1 and V2 are positive constants which are assumed to be related to each otheras V2 = γV1. So defined the parameter γ ≥ 1 is a measure of the depth of the periodicmodulation. The dynamics of the system is assumed to be governed by the DNLS equationof the form
idψn
dt= Vnψn − ε(ψn+1 + ψn−1) − |ψn|2ψn. (1)
This is the equation of motion for the Hamiltonian
H =∑
n
[
Vn|ψn|2 − ε(ψnψ∗n−1 + ψ∗
nψn−1) −1
2|ψn|4
]
, (2)
with canonical conjugated variables iψn, ψ∗n, and the implementation of periodic
boundary conditions through ψ0 = ψM , ψM+1 = ψ1. The star stands for complex conju-gation. In addition to the Hamiltonian H also the norm (excitation number N), definedby
N =∑
n
|ψn|2, (3)
is a conserved quantity for the system (1). Without loss of generality the coupling con-stant ε > 0 can be normalized to unity. This is realized by the rescaling transformationψn → √
εψn, Vn → εVn, and t→ t/ε. We therefore put ε = 1 unless the uncoupled limitat ε = 0 is referred to below.
3
We are interested in stationary solutions to (1) of the form
ψn(t) = ϕ(ω)n e−iωt (4)
with frequency ω and real-valued 1 amplitudes ϕ(ω)n . By linearizing Eq. (1) the dis-
persion relation for standing waves (phonons) with amplitudes of the form ϕ(ω)2k+1 =
ReCeiq(2k+1) and ϕ(ω)2k = ReDeiq(2k) for integers k = 0, 1, 2, · · · becomes
ω±
0 (q) = (V1 + V2)/2 ±√
(V2 − V1)2/4 + 4 cos2 q , (5)
where the allowed wave numbers q = 2πl/M for l = 0, 1, . . . ,M/2− 1 are determined bythe system size M . The ring must consist of an even number of oscillators in order toaccommodate the periodic boundary conditions. The four band edges ω±
0 (0) and ω±
0 (π/2)are all present for any finite system of the form M = 4m, where m is a positive integer.In particular, the frequencies V1 and V2 at the edges of the spectral gap (around thewave number q = π/2) correspond to periodic repetitions of the anti-phase oscillations(↑ 0 ↓ 0) and (0 ⇑ 0 ⇓), respectively, where the zeros indicate non-moving oscillatorsand the thickness of the arrows characterizes the relative size of the amplitude of theoscillations. Such anti-phase oscillations are absent for lattices of the type M = 4m+ 2.The other frequencies in the spectrum (5) are doubly degenerate.
The frequencies for nonlinear vibrations depend not only on the modulation param-eter γ = V2/V1, but also on the amplitude and, hence, implicitly on the norm and theHamiltonian. The “spectral” dependence H(N) of the system is, due to the fulfilmentof the relation ω = dH/dN for monochromatic oscillations, uniquely determined by thecharacteristic ω = ω(N).
When there is no spectral gap (γ = 1) it is possible to solve the equations exactlyfor M = 4 and obtain all the families of solutions, displaying a fascinating bifurcationstructure [2]. With a gap, γ > 1, the corresponding system reduces to four independentsets of two coupled oscillators with a total of 19 spectral curves ω(N), one of whichcan be found in the gap of the linear spectrum for modulation depths above a certaincritical value γC = 1 + 2
√2/V1 [22]. This is the first appearance of a solution that
can be thought of as an analogue of the gap soliton in a modulated lattice. Above acertain threshold value, this solution transforms smoothly to an out-gap soliton, andboth solutions constitute linearly stable excitations for large modulation depths γ [24].From the view point of the smallness of this system the concept of exponentially decayingtails for the gap soliton is not appreciated. In fact, these solutions are stabilized by thesystem size, which motivates a systematic study of the gap and out-gap solitons and theirstability properties in larger chains, letting the tails come into play.
1 As for the nonmodulated DNLS equation (γ = 1) in e.g. [2], there exist generally also stationary
solutions to Eq. (1) of the form (4) but with nontrivial complex amplitudes ϕ(ω)n (which cannot be
made real by a gauge transformation). Such solutions include, e.g., nonlinear Bloch waves with wavevector q, which also bifurcate from the linear phonon frequencies ω±
0 (q) given by (5) [29]. However, suchnontrivially complex stationary solutions cannot vanish at infinity for 1D DNLS models of the class (1)[30], and thus the localized stationary gap modes of interest in this paper must always appear with
real-valued ϕ(ω)n .
4
3. Linear stability analysis and bifurcations
The linear stability of a solution ψn(t) is analyzed by adding the perturbation
εn(t) = une−iΩt + w∗
neiΩ∗t (6)
to its time-independent amplitude ϕ(ω)n . The (complex) numbers un and wn are subject
to the same periodic boundary conditions as the solution and Ω is a (complex) eigen-
frequency of the linear eigenvalue problem obtained by linearization of (1) around thesolution (4). For the variables Un = un +wn and Wn = un −wn this eigenvalue problemcan be written (cf. Ref. [14,12])
0 L0
L1 0
Un
Wn
= Ω
Un
Wn
, (7)
where the self-adjoint operators L0 and L1 are defined by
L0Wn ≡[
Vn − ω − (ϕ(ω)n )2
]
Wn −Wn+1 −Wn−1, (8a)
L1Un ≡[
Vn − ω − 3(ϕ(ω)n )2
]
Un − Un+1 − Un−1. (8b)
Another common formulation of this problem is in terms of the eigenvalue Ω′ = iΩ,but here we choose to talk about eigenfrequencies Ω (being the physically more relevantconcept). Then linear stability of the stationary solution is equivalent to all eigenfrequen-cies Ω being real. The non-Hermitian problem (7) is infinitesimally symplectic, implyingthat if Ω is a solution, so is −Ω. Further, since Eq. (1) is invariant under global phaserotations, Ω = 0 is a doubly degenerate eigenfrequency for each stationary solution. Theassociated eigenvector, the phase mode, is proportional to the solution, which can be
expressed as Un = 0, Wn = ϕ(ω)n .
When the lattice size is a multiple of four, M = 4m, the phonons at the edges of thespectral gap exist as anharmonic solutions in the nonlinear regime, which are said toconstitute the boundaries of a nonlinear “gap” (see Fig. 1):
ωa(N) = V1 − 2N/M, (9a)
ωb(N) = V2 − 2N/M. (9b)
Solutions with such persistence are also termed nonlinear standing waves [14] with wavenumbers q inherited from the phonons in the limit of zero norm (|ψn|2 → 0). The firstsolution to bifurcate from the boundary (9b) is accompanied by an instability of thatsolution, allowing for an analytical treatment of the birth of the gap solitons by meansof linearizations of perturbations around the solution
ϕ(b)n = (0,
√B, 0,−
√B)m. (10)
The notation (· · · )m stands for m repetitions of the amplitudes within the parenthesisand B = V2 − ω. Defining A = V1 − ω, the width of the spectral gap is ∆ = V2 − V1 =(γ − 1)V1 = B − A. The gap width ∆ is chosen to vary the depth γ of the modulatedlattice, which will be measured in units of the potential V1 ≡ 4 throughout the study.
At every bifurcation point on the characteristic (9b), corresponding to (10), which is
represented by the phase mode in the meaning that L0ϕ(b)n = 0 in (8a), the operator
5
L1 should have a zero eigenfrequency Ω. This is realized by studying the spectrum ofeigenfrequencies of the problem (7). Taking the perturbation around the extended solu-tion (10) to be of the form U2k+1 = Ceiq(2k+1), U2k = Deiq(2k) and W2k+1 = Eeiq(2k+1),W2k = Feiq(2k) for integers k = 0, 1, 2, · · · yields the non-negative eigenfrequencies
Ω± =
√
A2 + 8 cos2 q
2±
√
(A2 + 8 cos2 q)2
4− (AB + 2 cos2 q)8 cos2 q , (11)
where the wave numbers q = 2πl/M for l = 0, 1, . . . ,M/2 − 1 are determined by thesystem size M = 4m. This implies that, whenever (AB + 2 cos2 q) < 0 in (11) thenonlinear gap boundary (9b) has an interval of modulational instability
|B − ∆/2| <√
(∆/2)2 − 2 cos2 q (12)
for gap widths ∆ large enough. At the boundaries of this instability interval the eigen-frequencies Ω− = 0 in Eq. (11) and the corresponding (real) frequencies are
ω±(q) = V ±√
(∆/2)2 − 2 cos2 q, (13)
where V ≡ (V1 + V2)/2 denotes the average on-site potential. The frequencies (13) areindeed the solutions of the eigenvalue problem L1Un = 0 for the subfields U2k+1 =Ceiq(2k+1) and U2k = Deiq(2k). The possible bifurcation points on (9b) thus appearexactly in the middle of the spectral gap at certain threshold gap widths
∆(l)M = 2
√2 |cos (2πl/M)| . (14)
For the specific integer l = M/4 the positive sign in (13) only gives the trivial solutionB = V2 − ω+(π/2) ≡ 0, whereas the negative sign truly results in a bifurcation point atA = V1 − ω−(π/2) = 0 for arbitrary ∆ > 0. This bifurcation is traced to the solution
ϕ(c)n = (
√A,
√B,−
√A,−
√B)m, satisfying ωc(N) = V − N/M , splitting off from
the nonlinear boundary (10). This is a generalization of the result for M = 4 in Ref. [22],and it can be understood in terms of a perturbation in the direction of the solution
ϕ(a)n = (
√A, 0,−
√A, 0)m, (15)
associated with the characteristic (9a). The choice of integers l = M/4 ± 1 ≡ m ± 1 foreach lattice size results in the smallest threshold values (14) – the critical gap widths
∆(m)C = 2
√
1 − cos(π/m) , m = 1, 2, 3, 4, . . . . (16)
As the gap parameter ∆ is increased above the threshold values (14) the pairs of bi-furcation frequencies (13) split and move in opposite directions on (9b). The birth ofthe solution with the highest bifurcation frequency should correspond to a solution withlowest Hamiltonian energy H for a given norm N in the gap. In fact, the existence of suchsolutions, and in particular a so-called “gap ground state”, has been proven [31,32]. Thisenergetically favourable state in the gap – the discrete gap soliton – is the bifurcatingsolution at the frequency ω+(π/2±π/(2m)) in (13), which appears on (9b) at the criticalgap width (16) for each lattice size M = 4m. The critical value (16) is essentially halved
([(∆(2m)C /2)2 − 1]2 = 1 − 1
2 (∆(m)C /2)2) as the integer m is doubled for large but finite
lattices. In the limit of an infinite lattice we get limm→∞ ∆(m)C = 0, although the actual
gap width ∆ must be positive to ensure the concept of a gap solution. In particular, the
6
2
2.5
3
3.5
4
4.5
5
5.5
6
0 5 10 15 20 25
ω
NFig. 1. Bifurcation diagram for M = 16 at ∆ = 2. The gap soliton bifurcates from the characteristic (9b)at the frequency ω+ ≈ 5.84, being rather close to the gap edge V2 = 6. The bifurcating solution at ω = 5is a bisoliton (two M = 8 gap solitons merged together). The bifurcation loop in the lower right corner,which will approach the lower gap edge V1 = 4 with increasing value of ∆, contains a solution similar tothe fundamental out-gap soliton but with a phase-shifted tail. The stable boundary (9a) is also shown.The stability regions of the boundary (9b) is marked by solid lines (cf. Fig 2).
limit limm→∞m∆(m)C = π
√2 can be taken as a lattice parameter for the existence of the
discrete gap soliton in the corresponding finite approximation, that is ∆ ≥ π√
2/m.We show in Fig. 2 the eigenfrequencies (11) (including −Ω±), as a function of frequency
ω for the solution (10) for m = 4 and ∆ = 2. For this choice of the gap width there is aninstability interval (12) present in the gap regime 4 ≡ V1 < ω < V1+∆ = 6 (0 < B < ∆),whose boundaries are given by the frequencies ω±(π/2 ± π/8) = 5 ± 2−1/4 ≈ 5 ± 0.84according to Eq. (13). At the highest frequency ω+ ≈ 5.84 the discrete gap solitonbifurcates from (10), and its characteristic ω(N) is depicted in Fig. 1. In a scenario wherethe parameter ∆ is increased from zero this instability develops on the characteristic (9b)
at ω = 5 for the critical gap width ∆(4)C = 2
√
1 − cos(π/4) ≈ 1.0824. The correspondingdependencies ω(N) and Ω(ω) are similar to those shown in Figs. 1 and 2 for the critical
value ∆(2)C = 2, but here the bifurcating solution at ω+(π/2± π/4) = V = 5 is a discrete
bisoliton, constituting two M = 8 gap solitons merged together. This is because M = 8
is a subsystem of the lattice M = 16. Of course, the critical value ∆(3)C =
√2 is absent
for this system since M = 12 is not a sublattice. More modulational instabilities developwhen the corresponding eigenfrequencies Ω−(ω) meet Ω = 0 (the phase mode) in the gapregime as ∆ is increased further. The smallest subsystem of this lattice is M = 4 and a
quadrusoliton (four M = 4 solitons) will be born at the critical gap width ∆(1)C = 2
√2.
An analogous formation of different types of multisolitons takes place in larger lattices.
7
-6
-4
-2
0
2
4
6
-2 -1 0 1 2 3 4 5 6
Ω
ω
Imaginary partReal part
Fig. 2. Eigenfrequencies Ω of Eq. (7) as a function of frequency ω for the solution (10) with M = 16
(m = 4) and ∆ = 2 = ∆(2)C
. Dark (grey dotted) lines represent real (imaginary) parts of Ω.
Below the spectral gap ω < V1 = 4 (in the out-gap regime 0 < A) the solution (10)suffers from oscillatory instabilities, corresponding to complex eigenfrequencies, of whichthe widest interval in Fig. 2 is associated with the sublattice M = 4. Oscillatory instabili-ties, manifesting resonances between internal modes, appear through Krein collisions [33],where eigenfrequencies leave the real axis as complex quadruplets. One may note thatthese collisions correspond to Hamiltonian Hopf bifurcations for the eigenvalue Ω′ = iΩ,see discussion in [34]. To each real eigenfrequency Ω > 0 of (7) one can associate a Kreinsignature [35,14] (assuming Un,Wn real)
K(Ω) = sign∑
n
UnWn , (17)
which can be interpreted as the sign of the (Hamiltonian) energy carried by the lineareigenmode [35]. The eigenfrequencies (11) are doubly degenerate, apart from the straightline Ω+(q = π/2) = A = V1 − ω and
Ω±(q = 0) =
√
A2 + 8
2±
√
(A2 + 8)2
4− 8(AB + 2) (18)
in Fig. 2, which are invariant for every lattice M = 4m. In the limit of an infinite latticethe continuous spectrum consists of two bands, Ω+ and Ω−. The Krein signatures ofthe eigenfrequencies to the left of the oscillatory instabilities in Fig. 2 are K(Ω±) = ±1.This can be shown by expanding Eq. (11) in the limit A → ∞. A Taylor expansionof (11) at small λ(q) ≡ AB + 2 cos2 q & 0 reveals a change in signature, K(Ω±) = ∓1,of the real eigenfrequencies Ω±(q, ω > V1) for q 6= π/2. An example of this change in
8
signature is given for the system M = 4 below. In particular, the internal modes Un andWn corresponding to Ω−(q) & 0 give information on the direction in which bifurcatingsolutions can be found. However, the mere fact that Ω−(ω) collides with Ω = 0 does notimply a bifurcating solution at ω, so its existence must be verified by other means.
35791113151 site
2
3
4
5
6
ω
-2-1012
amplitude
Fig. 3. The gap-out-gap discrete soliton for ∆ = 2 and M = 16 as a function of frequency ω. Note thatsite number 1 is moved around using the periodic boundary conditions for better vision of the tail.
Based on the proof of the existence of breathers [11], Newton iterative schemes are usedto numerically trace solutions to machine precision by means of a continuation versusthe model parameters from known solutions [36,37]. A parameter limit of the model isε = 0, where real stationary solutions of (1) have explicit analytical forms. Any solutionthat can be continued to this limit without bifurcations can be coded with a setting σof amplitudes ϕn ∈ 0,±
√A,±
√B, which depends on whether the frequency ω falls in
the gap or lies below the spectrum. This is because the anti-continuous limit ε/ω → 0 canbe reached either by putting ε = 0 or letting V1 −ω → ∞. The symmetric soliton, as theenergetically favourable gap state, can be continued to the limit ε → 0 leaving only onenon-zero amplitude, chosen as ϕ2 > 0, which is encoded σ0 = 0 ⇑ (0 O)2m−10 foreach lattice M = 4m. Here, bold symbols are used for the even amplitudes to distinguishthem from the odd ones, and the superscript 0 (∞) indicates how the anti-continuouslimit is reached. The solution for m = 4 and ∆ = 2 is shown in Fig. 3, where it bifurcatesfrom the nonlinear standing wave (10) with q = π/2 at ω+ ≈ 5.84. Following the smoothcharacteristic ω(N) in Fig. 1 down in frequency below V1 it becomes an out-gap solitonwith an ever increasing norm N . In the anti-continuous limit V1 − ω → ∞ the codingsequence of the symmetric (around the site number 2) discrete soliton in Fig. 3 is σ∞=↓ ⇑ ↓ O ↑ O ↓ O ↑ O ↑ O ↓ O ↑ O∞. One may note from Fig. 3 that the solution,being arbitrarily close to the nonlinear standing wave (10) at the bifurcation point, strivesfor exponential localization around the amplitude ϕ2 in the middle of the gap (ω = 5)
9
before it smoothly turns in the direction of the lower gap boundary (15) as it becomesan out-gap soliton (ω < 4). Note that, comparing with the nonlinear standing wave (15)there is always a phase defect associated with the soliton site n = 2 (the oscillations ofsites n = 1 and n = 3 are in-phase and not out-of-phase as for a pure wave with q = π/2),and to accommodate with the periodic boundary conditions an additional phase defect(around site n = 10) also has to be included. This failure of finding a tail with wavenumber q = π/2 will render the out-gap soliton in every lattice M = 4m unstable form ≥ 2. This argument also applies to the non-normalizable out-gap soliton in the limitm → ∞. An exception regarding soliton stability is found for the lattice M = 4 [24],which is treated as a special case below.
The corresponding analysis for the class of lattices, M = 4m+2, lacking the nonlinearstanding wave (10), fully relies on numerical methods. The spectral gap edges in this caseapproach the values V1 and V2 only asymptotically as m → ∞. The appearance of thegap soliton in such a finite lattice can be divided into two cases from which the infinitelattice is approached in a different way as compared to the class of lattices M = 4m.
3.1. Case I: M = 4
The internal modes corresponding to the eigenfrequency Ω−(q = 0) & 0 in Eq. (18) areuseful in finding the gap soliton that bifurcates from the nonlinear standing wave (10)
for ∆ > 2√
2 = ∆(1)C in the lattice M = 4. This eigenfrequency can be expanded as
Ω− ≈ 2√
2 + ∆√
2/A close to the limit A→ ∞ (cf. Fig. 2), resulting in an eigenvector
Un mod2
Wn mod2
=
0
2/A
,
−√
2/A
1 −√
2/A
+ O(1/A2) (19)
with signature K(Ω−) = −1. Taking λ(0) = AB + 2 & 0 to be a small parameter theeigenmode associated with the eigenfrequency Ω− ≈
√8λ/
√A2 + 8 can be expressed as
Un mod 2
Wn mod 2
=
−B
−√
2λ√A2 + 8
,
1
(B + ∆)√λ
√
2(A2 + 8)
+ O(λ), (20)
with K(Ω−) = +1. Hence, the Krein signature of the eigenfrequency Ω− is changed(through Krein collisions with Ω+) at the boundaries of the oscillatory instability domainin Fig. 2. The eigenmode (19) has the natural interpretation as vibrations in the even-site-sublattice in the decoupled limit. At the bifurcation point λ = 0, the exact solution (10)is thus to be perturbed in the direction U1 = U3 = −BU2, U2 = U4
√B in tracing the
gap soliton characteristic caught in Fig. 4 at ω+(0) ≈ 5.95. The slope of this characteristicis seen to initially fulfil ∂N/∂ω > −2 = ∂N/∂ωb(N), and the soliton tends to localize witha frequency ω in the gap. When 2
√2 < ∆ < 3 this stable gap soliton bifurcates with the
unstable solution born at the other bifurcation point on the nonlinear gap boundary (9b)at ω−(0) ≈ 5.02 in Fig. 4. The stable gap soliton (0 ⇑ 0 O0) is doubly degenerateso that its split off from the nonlinear boundary at ω+(0) classifies as a supercriticalpitchfork bifurcation. (The unstable split off at ω−(0) ≈ 5.02 in Fig. 4 is a subcriticalpitchfork bifurcation.) This creation can be thought of as a resonance phenomenon withthe bifurcation loop (containing solutions with a similar type of oscillation) appearing
10
4
4.5
5
5.5
6
2 3 4 5 6 7
ω
N
bd
e
UnstableStable
Fig. 4. Bifurcation diagram for M = 4 at ∆ = 2.98 > 2√
2 = ∆(1)C
. Solid (grey dotted) lines representstable (unstable) solutions. The large dots are bifurcation points. The boundary (9b) is denoted by b.
in the lower right corner of Fig. 4, which bifurcate with each other for 0 < ∆ < 3. Inthe anti-continuous limit the unstable solution is labelled σd∞ = ↓ ⇑ ↓ ⇓∞, whereasthe stable solution has the code σe∞ = ↓ ⇑ ↓ O∞. These four solutions all cometogether at a bifurcation point (N,ω) = (5.5, 4.5) for the numerically exact value ∆∗ =(γ∗−1)V1 = 3. [22,24]. When ∆ is increased above ∆∗ the solutions change partner in themeaning that the two unstable solutions constitute a continuous solution ωd(N) and thecharacteristic of the gap soliton is now smoothly connected to that of the out-gap soliton,ωe(N). This bifurcation, connecting the gap and out-gap solitons for ∆ = 3, is illustratedin Fig. 5, where the collisions of real eigenfrequencies Ω with Ω = 0 (the phase-mode)meet at ω = 4.5. The oscillatory instability shown in Fig. 5 results from resonancesbetween the internal modes corresponding to the other two (positive) eigenfrequencies.With a further increase of ∆ > 3, the window of oscillatory instability, originally locatedin the frequency region (−1.98 < ω < −0.26) of the out-gap soliton, shrinks and movesinto the gap regime where it finally disappears for the gap width ∆ = 6 [24].
3.2. Case II: M = 4m, m = 2, 3, 4, . . .
The modulational instability (12) associated with m = 1 remains also for m ≥ 2 fromwhich a solution in the form of a soliton train, obtained by merging discrete solitons
0 ⇑ 0 O0 together one by one in phase, bifurcates at the critical gap width ∆(1)C . This
extended type of solution in the gap connects with the corresponding out-gap multisoliton(↓ ⇑ ↓ O)m∞ via a bifurcation analogous to the one shown in Fig 4 and Fig. 5, but
11
-10
-5
0
5
10
-4 -2 0 2 4 6
Ω
ω
Imaginary partReal part
Fig. 5. Eigenfrequencies Ω versus frequency ω for the gap and out-gap discrete solitons at∆∗ = (γ∗ − 1)V1 = 3. The two stable (besides the oscillatory instability −1.98 < ω < −0.26) branchesjoin in a bifurcation point at ω = 4.5. Solid (grey dotted) lines represent real (imaginary) parts of Ω.
these excitations no longer constitute linearly stable solutions.In regards to extended solutions and their linear stability, the configuration obtained
by decorating the entire lattice with an even number of solitons 0 ⇑ 0 O0 in anti-phasealso yields a solution in the gap of the linear spectrum. It is smoothly connected to thecorresponding out-gap solution, of which the simplest is ↓ ⇑ ↓ O ↑ ⇓ ↑ O∞. Numericalcalculations reveal no strong instabilities for this particular solution, neither for frequen-cies in the out-gap region nor in the gap, but weak instabilities of oscillatory type arepresent. For larger lattices, oscillatory instabilities (quite generally increasing in numberdue to more possible resonances between the internal modes) associated with localizedsolutions become weaker, and they should vanish completely in the limit of an infinitesystem [38]. This reasoning applies to the continuous bands, while resonances involvingthe localized eigenmodes still can produce instabilities [12]. Oscillatory instabilities oflocalized solutions that weaken with the system size are discarded in the following. Foran extended solution on the other hand one cannot guarantee that oscillatory instabilitiesdisappear as m→ ∞, which is perfectly clear from Eq. (12) and Fig. 2.
The configuration ↓ ⇑ ↓ O ↑ ⇓ ↑ O∞ defines an antisymmetric soliton. It does notbifurcate from the solution (10) for m = 2, but can be continued to a frequency slightlyabove V2 in the linear limit. The solution with the lowest energy in the gap for m = 2 ishowever the discrete soliton 0 ⇑ (0 O)30. Whenever this solution exists it is smoothlyconnected to the corresponding out-gap soliton σg∞ = ↓ ⇑ ↓ O ↑ O ↑ O∞, whichbifurcates with the unstable solution σf∞ = ↓ ⇑ ↓ ⇓ 0 ⇑ 0 ⇓∞ for 1.20 . ∆ ≤ 2
as shown in Fig. 6. At the critical gap width ∆(2)C = 2 the unstable solution σf∞
12
4.8
4.9
5
5.1
5.2
5.3
3 3.5 4 4.5
ω
N
g bf
Boundary
Fig. 6. Bifurcation diagram for M = 8 at ∆ = 2 = ∆(2)C
. Solid (dotted) lines represent stable (unstable)solutions. The large dot marks the bifurcation point on the boundary (9b), denoted by b.
resonates with the nonlinear gap boundary, resulting in a modulational instability (12)on the solution (10) for ∆ > 2. Since the slope ∂N/∂ωf < −4 = ∂N/∂ωb(N) near thebifurcation point ω+(π/2 ± π/4) = 5, we may say that the gap soliton ωg “borrows” apart of the unstable solution ωf in connecting to the nonlinear gap boundary (9b) via asubcritical (inverse) pitchfork bifurcation. The change of stability for the gap soliton takesplace at a saddle-node bifurcation, which is the turning point where ∂N/∂ω changes signin Fig. 6. The unstable part becomes smaller for larger values of ∆ but according to ournumerics it does not seem to disappear completely. The instability of the gap soliton whenpassing the turning point is thus of the Vakhitov-Kolokolov type [39], which is found tobe an invariant feature of the gap soliton for finite m > 2, although the instability regionshrinks as m increases. In the infinite-size limit the change of stability is not possible to
resolve numerically since the critical gap width ∆(m)C , associated with the birth of the
gap soliton 0 ⇑ (0 O)2m−10 from the characteristic (9b), approaches zero as m→ ∞.The bifurcation loop involving the out-gap soliton for m = 3 is a smooth parabola (like
the one form = 2 in Fig. 6) for small gap widths ∆, but the unstable solution deforms intoa multivalued characteristic ω(N) as shown in Fig. 7 when the gap parameter approaches
the critical value ∆(3)C =
√2 from below. This is a precursor of the development of two
new solutions and a second bifurcation involved in connecting the fundamental out-gap soliton to the nonlinear gap boundary for all m ≥ 4. For m = 4 this bifurcationwill take place between the solutions marked by open circles and filled triangles around(ω,N) ≈ (4.2, 6.9) in Fig. 8 as ∆ is increased further above the corresponding critical gapwidth. After this point the soliton (plotted with filled circles) uses a part of the open-
13
4.45
4.5
4.55
4.6
4.65
4.7
4.75
4.2 4.4 4.6 4.8 5 5.2 5.4 5.6
ω
N
b
Boundary
Fig. 7. Bifurcation diagram for M = 12 at ∆ = 1.4142 <√
2 = ∆(3)C
. Solid (dotted) lines representstable (unstable) branches of the (out-)gap soliton ↓⇑ ↓ O ↑ O ↓ O ↓ O ↑ O∞ and the solution0 ⇑ 0 ⇓ ↑⇑ 0 ⇓ 0 ⇑ ↑⇓∞ coming from above the nonlinear gap boundary b.
circle solution as an intermediate solution in connecting to the filled-triangle solution ina picture that resembles the multivalued function ω(N) of Fig. 7. For larger modulationdepths the intermediate open-circle solution disappears so that the soliton connects tothe filled-triangle solution in a monotonous curve as in Fig. 1. However, a very smallpart of the unstable open-triangle solution (corresponding to the subcritical pitchforkbifurcation in the left inset of Fig. 8) seems to remain also for ∆ large according to ournumerical simulations. Hence, from this hypothesis, the gap soliton undergoes a Vakhitov-Kolokolov type of stability change in a saddle-node bifurcation very close to the nonlineargap boundary (9b). This is the case for the bifurcating gap soliton at ω+ ≈ 5.84 in Fig. 1.
3.3. Case III: M = 4m+ 2, m = 2, 3, 4, . . .
For the class of lattices M = 4m+ 2 the solution (10) does not exist, but
σ∞ = 0 ⇑ 0 ⇓ (↑ O ↓ O)m−1 ↑ ⇓∞ (21)
instead plays the role of a nonlinear gap boundary for ∆ small enough in the meaning thatit is continuable to the phonon at the gap edge ω+
0 (π/2 +π/M) of the spectrum (5). Forgap parameters larger than some critical gap width, dependent upon M , the solution (21)can no longer be continued to the gap edge in the limit of zero norm N , but bifurcateswith the solution
σ∞ = ↓ ⇑ ↓ ⇓ (↑ O ↓ O)m−1 ↑ ⇓∞, (22)
14
3.9
4.1
4.3
4.5
4 6 8 10
ω
N
BoundariesSoliton
4.541
4.543
4.31 4.33
2
3
4
5
5 15 25 35
ω
N
Fig. 8. Bifurcation diagram for M = 16 at ∆ = 1.0824 > 2√
1 − 1/√
2 = ∆(4)C
. The left inset showsan enlargement of the resonance between the solution denoted by open triangles, the nonlinear gapboundary (plotted with thin lines) and the solution represented by filled triangles. The large dots denotethe almost overlapping bifurcation points. The filled circles constitute the soliton characteristic, whichbifurcates with the solution marked by open circles. As ∆ is increased there will be a change of partnerbetween the filled-triangle solution and the open-circle solution in a bifurcation at (ω, N) ≈ (4.2, 6.9),leaving a picture similar to Fig. 7 for the gap-soliton. The right inset shows a wider perspective of thebifurcation diagram, but with the open-triangle solution 0 ⇑ 0 ⇓ ↑⇑ 0 ⇓ 0 ⇑ 0 ⇓ 0 ⇑ ↑⇓∞ removed.We do not state the codes of the other two solutions because they will change in bifurcations well belowthe spectral gap (around ω ≈ 1.75 in the right inset) as the gap parameter is increased.
which is depicted with dotted lines in Fig. 9 for m = 2. For gap modulations smallerthan the critical gap width the unstable solution (22) bifurcates with the out-gap soliton
σ∞ = ↓ ⇑ ↓ O (↑ O ↓ O)m−1 ↑ O∞. (23)
At the critical gap width these three solutions all come together in a bifurcation point.We refer to this bifurcation as the defining moment for the discrete gap soliton, since theout-gap soliton (23) is connected with the solution that is continuable to the gap edge inthe linear limit beyond this point. The gap-out-gap soliton characteristic for a gap width∆ = 0.84 slightly larger than the critical value is plotted with dark solid lines in Fig. 9for m = 2. Its connection with the solution (plotted with thin lines) that is continuableto the gap edge ω+
0 (3π/5) ≈ 5.17 in the linear limit involves two saddle-node bifurcationsat which ∂N/∂ω = 0. However, as ∆ is increased the characteristic ω(N) for the solitonbecomes a monotone function smoothly continuable to the upper gap edge. This scenariois invariant with respect to m ≥ 2. In fact, there is a similar picture of such a bifurcationin Ref. [12] (Fig. 6 in Sect. 5.2) for m = 60. The splitting of the solution (21) in Fig. 9 issimilar to that of the filled-triangle solution in bifurcating with the open-circle solution
15
3
3.5
4
4.5
5
5.5
0 2 4 6 8 10 12
ω
NFig. 9. Bifurcation diagram at ∆ = 0.84. The nonlinear gap boundary (21) for m = 2 (thin lines) is nolonger continuable to the gap edge, but bifurcates with the unstable solution (22) (dotted lines). Thesoliton characteristic for M = 10 (dark solid lines) connects via saddle-node bifurcations to the solution(a former part of the thin-line boundary (21)) that is continuable to the gap edge ω+
0 (3π/5) ≈ 5.17 in the
linear limit. The soliton characteristic for M = 6, starting at the corresponding gap edge ω+0 (2π/3) ≈ 5.50
for ∆ = 0.84, is plotted with grey solid lines in the same diagram.
as ∆ is increased in Fig. 8, being a representative for the class of lattices M = 4m. Alsothe critical value at which this takes place tends to zero in the limit m→ ∞. It is clearfrom the code (23) that the tail with wave number q = π/2 for the fundamental solitonperfectly matches the class of lattices M = 4m + 2. This is in contrast to the Case II,where a mismatch in the tails renders the out-gap soliton unstable.
3.4. Case IV: M = 6
The lattice M = 6 is somewhat of a special case. Without a gap (monoatomic lattice)the solution (23), which will become the gap soliton for some critical value ∆ > 0, form = 1 is identical to the solution (21), where (· · · )0 should be removed. These solutionsbifurcate with the degenerate solution (22), which in turn can be continued to zero norm(cf. Ref. [40]). In this case the splitting of (21) happens for arbitrarily small ∆ > 0, andthe dependence ω(N) for the (out-)gap soliton (23) becomes directly a monotone functioncontinuable to the upper gap edge ω+
0 (2π/3) in Eq. (5). This soliton characteristic is alsodepicted in Fig. 9 for ∆ = 0.84.
16
4. Dynamics and statistical mechanics
Let us now discuss the statistical nature of the asymptotic dynamics in the thermo-dynamic limit, M → ∞, along the lines described in, e.g., [25–28]. For convenience, weredefine the Hamiltonian (2) as H′ = −
(
H− V N)
, where V is the average on-site po-tential as in (13). For the DNLS model without on-site potential (Vn ≡ 0), there is aphase transition at the curve [25]
〈H′〉M
=
( 〈N〉M
)2
. (24)
Below this line the system is expected to thermalize according to the Gibbsian equilibriumdistribution with temperature T = β−1 (in units of Boltzmann’s constant kB ≡ 1)and chemical potential µ for typical initial conditions. On the other side, the systemshows “negative temperature”-type behaviour and statistical localization in terms ofspontaneous formation of persisting breathers (intrinsic localized modes) is typicallyfound [25–28]. Direct calculation of the nonlinear gap boundaries (9) with wave numbersq = π/2 gives the dependencies (plotted in Fig. 10)
H′a,b
M= (+,−)
∆
2
(
N
M
)
+
(
N
M
)2
(25)
of the Hamiltonian and norm densities, respectively. The corresponding phase transition
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5
Ham
ilton
ian
norm
0π /2
out-gap soliton
phase line
gap soliton
π /2π
Fig. 10. Hamiltonian H′/M versus norm N/M densities for solutions with wave numbers q ∈ 0, π/2, πand ε = 1 corresponding to the band edges continued into the nonlinear regime and the gap bound-aries (25). The gap-out-gap soliton is for clarity depicted for the modest lattice M = 32.
17
line, defined by the boundary (β = 0, µ = ∞ at constant βµ) is here expected to fall in-between the curves (25). In deriving this conjecture it is suitable to perform a canonicaltransformation to action-angle variables, ψn =
√Ine
iφn , taking the Hamiltonian H′ into
H′ =
M∑
n=1
[
(
V − Vn
)
In + 2ε√
InIn−1 cos (φn − φn−1) +1
2I2n
]
. (26)
The prediction of macroscopic averages in the thermodynamic limit can be made fromthe grand-canonical partition function
Z =
∫ ∞
0
∫ 2π
0
M∏
n=1
dφndIne−β(H′+µN) , (27)
where the conserved norm N =∑M
n=1 In is treated in analogy with the “number ofparticles”. Considering a general partition M = Ma + Mb of on-site modulations Va,b
(not necessarily periodic) and integrating over the phase variables φn yields
Z(β, µ) = (2π)M
∫ ∞
0
M=Ma+Mb∏
n=1
dInI0(2βε√
InIn−1) e−βIn(V −Va,b+In/2+µ), (28)
where I0(α) = (1/π)∫ π
0 exp(α cos θ) dθ is the modified Bessel function of the first kind(not to be confused with the action I = N). In the high-temperature limit, β → 0+,I0 can be approximated by one, which also is consistent with the fact that an ensembleof independent thermalized units mathematically corresponds to letting ε → 0. Theremaining integral then factorizes into a product of independent integrals for each ofthe two sublattices, which can be evaluated in the spirit of Refs. [25,28] in the high-temperature limit (β → 0+, µ→ ∞ at constant βµ). The result is
Z ' (2π)M 1
(βµ)M
[
1 − β(V − Va)
βµ− β
(βµ)2
]Ma [
1 − β(V − Vb)
βµ− β
(βµ)2
]Mb
, (29)
and
lnZ 'M ln(2π) −M ln(βµ) −[
MV − (MaVa +MbVb)] β
βµ−M
β
(βµ)2. (30)
From this, we obtain the averaged energy density
〈H′〉M
=1
M
(
µ
β
∂
∂µ− ∂
∂β
)
lnZ '(
V − MaVa +MbVb
M
) 〈N〉M
+
( 〈N〉M
)2
, (31)
using the average excitation norm density
〈N〉M
= − 1
β
∂ lnZ∂µ
1
M' 1
βµ. (32)
Thus, the phase transition curve (31) with infinite temperature (β = 0) falls in-betweenthe dependencies (25) of the nonlinear gap boundaries for a periodic modulation satisfyingMa = Mb = M/2 (see Fig. 10), and it formally agrees with Eq. (24). It could be worthmentioning the possibility of tuning the position of the curve (31) for models with morethan one fundamental spectral gap.
Although we did not prove it, we conjecture that the Hamiltonian H′ is bounded frombelow by the plane wave σπ∞ = (0 ⇑)m∞ with q = π that can be smoothly continued
18
0100
200300
400500
site
50000
100000
150000
200000
time0
0.4
0.8|ψn|2
Fig. 11. Instability-induced dynamics of the gap boundary (10) for ∆ = 0.2 > ∆(128)C
and M = 512.
to the upper band edge phonon in the linear limit. This means that the curve H′π/M
in Fig. 10 presumably bounds the accessible region in phase-space and thus defines zerotemperature (β = ∞). In the range 0 < β < ∞ of parameter space, where the systemis expected to thermalize in the (normal) Gibbsian sense, we first focus our attentionon the gap boundary H′
b/M corresponding to Eq. (25) with a negative sign for the firstterm. Because of the intervals (12) modulational instabilities [13] of the gap boundary are
expected. In fact, by tuning the gap width ∆ > ∆(m)C for a given lattice size M one can
essentially determine how many gap solitons that will appear through the modulationalinstability. Numerical integration of Eq. (1) for the gap boundary (10) is shown in Fig. 11,where three metastable solitons are born from unstable modulations along the lattice.The solitons begin to move with respect to each other and interact in the manner ofdiscrete solitons. This locally coherent translational motion, characterised by inelasticsoliton-soliton and soliton-phonon scattering, typically coalesces into the lowest energy(i.e., smallest H, or equivalently largest H′) state in the gap – the discrete gap soliton.The coherence is gradually lost and the motion enters a space-time chaotic regime wherethe system finally approaches a state with equipartition of energy (not shown in Fig. 11).
A statistical treatment above the curve (24) can be made by formally introducingnegative temperatures (β < 0), but the Hamiltonian H′ is not bounded from above sothat the description breaks down in the thermodynamic limit. The state with the largestH′ (or lowest energy H) at a given N is the ordinary discrete soliton ↑ O (0 O)m−1∞,which bifurcates from the solution σ0∞ = (↑ O)m∞ with wave number q = 0depicted in Fig. 10. Also from this solution modulational instabilities give rise to discrete
19
2040
6080
100120
site
0
200000
400000
600000
800000
100000
1200000
time0
1.53.0
|ψn|2
(a)
2040
6080
100120
site
0
200000
400000
600000
800000
1000000
1200000
time06
12|ψn|2
(b)
Fig. 12. Instability-induced dynamics of the gap-out-gap solution (23) for ∆ = 2 and M = 130. For thefrequency ω = 4.9 in the gap regime (a) and below the linear spectrum in the out-gap regime (b).
20
solitons (breathers), but they (or preferably one of them) persist(s) in time. Heuristicarguments for such a behaviour can be made in the micro-canonical ensemble, where thecreation of localized excitations is statistically favourable for negative temperatures; theytend to cool the system off by creating hot spots of localized energy [25,27,28].
The most interesting feature here is realized by the out-gap soliton crossing the phasetransition curve as it becomes a gap soliton as depicted in Fig. 10. The instability-induceddynamics of this solution is shown in Fig. 12, where the localized gap soliton exhibitsthe characteristic behaviour of the thermalization regime in the long-time limit, whilepersisting localized breathers are created from the original extended out-gap soliton. Thedynamics in Fig. 12 result from truncations of the numerically exact solutions in regionswhere they are oscillatorily unstable. It should be noted that there are indeed values ofthe frequency for which the gap soliton is linearly stable and actually remains in such acoherent state for very long integration times also under small perturbations.
The phase transition of the gap-out-gap soliton is illustrated for the modest systemM = 32 in Fig. 10, but this curve asymptotically approaches the nonlinear gap boundaryH′
a/M with increasing size, while the gap soliton tends to bifurcate from the other gapedge. This implies that the phase transition does remain also in the limit of an infinitesystem. It is interesting to note that the out-gap soliton with phase-shifted tail, obtainedby shifting the codes in the parenthesis in (23) to (↓ O ↑ O)m−1, seems to remain inthe regime where localization is statistically favourable. The corresponding phase-shiftedout-gap soliton for the system M = 16 is shown in Fig. 1, where the bifurcation loopwill approach the gap edge V1 = 4 from below with increasing value of ∆. This suggestthat the phase transition of the gap-out-gap solution takes place around the frequencyω(N) = V1 for large modulation depths.
5. Summary
This study was concerned with time-periodic solutions of the DNLS equation withcubic nonlinearity and a periodic on-site potential, modelling arrays of coupled nonlinearoptical waveguides, photonic crystals or Bose-Einstein condensates in periodic potentials.
For finite lattices supporting a nonlinear gap boundary with wave number q = π/2the bifurcation frequencies giving rise to solutions in the gap with localization rangingfrom extended “soliton trains” to the exponentially localized discrete soliton were foundin analytical form. The emergence of the gap soliton via pitchfork bifurcations from thenonlinear gap boundary was found to be generic for the class of lattices M = 4m. Thisappearance, accompanied by modulational instabilities of the nonlinear gap boundary,was shown to be associated with critical gap widths, turning to zero in the infinite-sizelimit. Despite the fact that tangency of the soliton is to be expected at gap edge, asaddle-node bifurcation in the vicinity of the nonlinear gap boundary might well remainas the gap width tends to infinity, except for the lattice M = 4. The fundamental gapsoliton was found to turn unstable as it smoothly transforms into an out-gap soliton forfrequencies outside the spectral gap due to a mismatch in the extended tails.
For the class of lattices M = 4m+ 2, where the tails of the symmetric soliton match,the gap boundary only asymptotically approaches a standing wave with q = π/2. Thissolution was found to split in a bifurcation scenario at critical gap widths, leading to thesoliton being a continuable orbit to the gap edge in the linear limit. Here the saddle-node
21
bifurcation was found to disappear for large modulations depths. The critical gap widths,associated with the birth of the discrete soliton, were numerically found to approach zeroin the infinite-size limit. The lattice M = 6 classifies as a special case in these regards.It could be of future interest to study the continuum limit of this model.
The tunable instability of the nonlinear gap boundary, as a result of the bifurcatinggap soliton, was numerically shown to excite metastable soliton states which graduallylost their coherence and the system approached thermal equilibrium in the long-timelimit. The gap soliton, belonging to the thermalization regime, was shown to undergo aphase transition to the out-gap regime, where the spontaneous formation of persistingdiscrete breathers was numerically found in accordance with statistical predictions.
Acknowledgements
In memory of Prof. Rolf Riklund, for his continuous support and encouraging dis-cussions. We thank M.M. Bogdan, A.V. Gorbach, and O.V. Usatenko for discussions.Financial support from the Swedish Research Council, the Royal Swedish Academy ofSciences, and the Swedish Institute is gratefully acknowledged.
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