PHYSICAL REVIEW E 74, 026601 共2006兲

Gap solitons in quasiperiodic optical lattices

Hidetsugu Sakaguchi1 and Boris A. Malomed2

1
Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences,
Kyushu University, Kasuga, Fukuoka 816-8580, Japan
2
Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
共Received 5 March 2006; revised manuscript received 25 May 2006; published 8 August 2006兲
Families of solitons in one- and two-dimensional 共1D and 2D兲 Gross-Pitaevskii equations with the repulsive
nonlinearity and a potential of the quasicrystallic type are constructed 共in the 2D case, the potential corresponds
to a fivefold optical lattice兲. Stable 1D solitons in the weak potential are explicitly found in three band gaps.
These solitons are mobile, and they collide elastically. Many species of tightly bound 1D solitons are found in
the strong potential, both stable and unstable 共unstable ones transform themselves into asymmetric breathers兲.
In the 2D model, families of both fundamental and vortical solitons are found and are shown to be stable.

DOI: 10.1103/PhysRevE.74.026601 PACS number共s兲: 42.65.Tg, 03.75.Lm, 61.44.Br, 42.70.Qs

I. INTRODUCTION which, essentially, treats the extended state as a segment of a

nonlinear Bloch wave, bounded by two fronts 共domain walls兲
Solitons in Bose-Einstein condensates 共BECs兲 are cur- that are sustained by the strong OL 共a similar wall between
rently a subject of intensive theoretical and experimental filled and empty domains was predicted in BEC with self-
studies. Solitons supported by weak attractive interactions attraction in Ref. 关11兴兲.
between atoms were created in the condensate of 7Li trapped Stable GSs were also predicted in 2D and 3D settings 关5兴,
in strongly elongated 关nearly one-dimensional 共1D兲兴 traps 关1兴 including 2D solitons with embedded vorticity 关12,13兴. In
共although the actual shape of the solitons was actually nearly nonlinear optics, similar 2D spatial solitons 关14兴 and 2D lo-
three-dimensional, rather than nearly 1D; the latter feature calized vortices 关15兴 were predicted in photonic crystals and
was observed in a salient form in the recent experiment 关2兴, photonic-crystal fibers, as well as in periodic photonic lat-
where solitons were created as a residual pattern after dy- tices induced by perpendicular laser beams in photorefractive
materials 关16兴. In media of the latter type, both fundamental
namical collapse took place in the 85Rb condensate兲. The
关17兴 and vortical 2D solitons have been created in the experi-
stability of the solitons, with a given number of atoms,
ment 关18兴. A difference from the self-repulsive BEC is the
against collapse is secured, in these settings, by a combina- self-focusing character of the nonlinearity in nonlinear optics
tion of the trap’s geometry and small absolute value of the 共which is cubic in photonic crystals and saturating in photo-
negative scattering length characterizing the attraction be- refractive media兲; for this reason, the optical solitons usually
tween atoms 共⬃0.1 nm in the case of 7Li, but up to ⬃2 nm belong to the semi-infinite band gap, where the effective
in the condensate of 85Rb兲. A very accurate description of the mass is always positive 共although vortex solitons in a finite
solitons is provided by the mean-field approximation based gap were also created in a photorefractive medium 关19兴兲.
on the Gross-Pitaevskii equation 共GPE兲 关3兴. The GPE with self-attraction and periodic 共optical-lattice兲
Positive scattering length, which corresponds to repulsive potential gives rise to 2D solitons and vortices 关20兴 similar to
interactions, is more generic in BEC 共in the above-mentioned those found in the above-mentioned optical models. More-
experiments, the negative scattering length was actually arti- over, it has been recently demonstrated that both the GPE
ficially induced by means of the Feshbach resonance, both in with the self-focusing cubic term and OL potential, and its
7
Li and 85Rb兲. In a repulsive condensate, gap solitons 共GSs兲 counterpart with the saturable nonlinearity, that pertains to
may be created as a result of the interplay of the self- photorefractives, give rise to stable higher-order localized
defocusing nonlinearity and periodic potential induced by an vortices 共which are built as rings of unitary vortices兲 and
optical lattice 共OL, i.e., an interference pattern created by supervortices 共similar rings with global vorticity imprinted
counterpropagating laser beams illuminating the condensate兲 onto them兲 关21兴.
关4,5兴. GSs emerge in band gaps of the system’s linear spec- Quasiperiodic OLs can be easily created too: in the 1D
trum, since the combination of a negative effective mass, case, as a superposition of two sublattices with incommen-
appearing in a part of the bands adjacent to the gaps, with the surate periods, and in the 2D case, as a combination of
repulsive interaction is exactly what is needed to create a N = 5 or N 艌 7 quasi-1D sublattices with wave vectors k共n兲 of
soliton. Theoretical models for GSs in BEC were reviewed in equal lengths, which make equal angles 2␲ / N between
Ref. 关6兴, and rigorous stability analysis for them in the 1D themselves. In particular, the 2D lattice with N = 5 is known
case was developed in Ref. 关7兴. Creation of a GS in the as the Penrose tiling 共PT兲. The band-gap spectrum of 2D
experiment was reported in Ref. 关8兴, in 87Rb condensate photonic crystals of the PT type has been studied in detail
placed in a quasi-1D trap supplemented by a longitudinal 关22兴, with a conclusion that true 共omnidirectional兲 band gaps
OL; the soliton contained a few hundred atoms. In a subse- may be supported by the PT.
quent experiment, large-size confined states with much larger The use of quasiperiodic lattices, in one and two dimen-
numbers of atoms were discovered in a stronger OL 关9兴; an sions alike, offers new degrees of freedom that allow one to
explanation of this observation was recently proposed 关10兴, engineer desirable band gaps in the spectrum and, in this

1539-3755/2006/74共2兲/026601共7兲 026601-1 ©2006 The American Physical Society

HIDETSUGU SAKAGUCHI AND BORIS A. MALOMED PHYSICAL REVIEW E 74, 026601 共2006兲

FIG. 1. A part of the band-gap structure in the

linear version of Eq. 共1兲. Crosses in 共b兲 indicate
values of ␮ for three GSs shown in Fig. 2.

way, design soliton families in nonlinear media. Several the- II. ONE-DIMENSIONAL SOLITONS
oretical works sought for solitons in models combining qua-
In the mean-field approximation, the evolution of the
siperiodic lattice potentials and cubic self-focusing. In an
single-atom function ␾ obeys the GPE with the repulsive
early work 关23兴, solitons were not found in a 1D quasiperi-
nonlinear term and quasiperiodic potential U共x兲. In the nor-
odic model; however, they were later discovered in a “deter-
ministic aperiodic” discrete nonlinear Schrödinger 共NLS兲 malized form, the equation is
equation, which may be regarded as a limit case of the 1D ⳵␾ 1 ⳵ 2␾
model with a very strong quasiperiodic potential 关24兴. Local- i =− + 兩␾兩2␾ + U共x兲␾ . 共1兲
ized and delocalized solutions were also studied in the 1D ⳵t 2 ⳵x2
model with a superlattice 共i.e., an ordinary OL subjected to As said above, the 1D potential is a combination of two
an additional modulation, which is different from a quasic- incommensurate spatial harmonics with equal amplitudes ␧,
rystal兲 关25兴. Recently, 2D solitons 共with zero vorticity兲 were
numerically constructed in a model of a photonic crystal U共x兲 = − ␧兵cos关␲共x − L/2兲兴 + cos关q␲共x − L/2兲兴其, 共2兲
made of a self-focusing material, with N = 12, in terms of the
above definition 关26兴. where x = L / 2 is the central point of a large 共L Ⰷ 1兲 trapping
The objective of this work is to find GSs supported by the domain, 0 ⬍ x ⬍ L, the period of the first sublattice is normal-
interplay of the cubic repulsive nonlinearity and quasiperi- ized to be 2, and q is an irrational number. Below, we display
odic lattice potentials. We will report systematic results for results for q = 共冑5 + 1兲 / 2 ⬇ 1.62.
fundamental solitons in the 1D and 2D models 共the latter one As is known, an exact band-gap spectrum of the linear
will be elaborated for the PT lattice兲. In the 1D case, loosely Schrödinger equation with a quasiperiodic potential is frac-
bound GSs in the weak lattice are mobile. Tightly bound GSs tal. Without the aim to display the spectrum in full detail, in
in the strong lattice are found in many modifications, some Fig. 1 we show the part that is relevant to the quest for gap
stable and some not. Stable vortex solitons in the PT poten- solitons. Bands of values of chemical potential ␮ corre-
tial will be demonstrated too. sponding to families of quasi-Bloch states,

FIG. 2. Examples of stable gap solitons in the

1D model for ␧ = 1, with the chemical potential
and norm ␮ = 0.658, N = 0.104 共a兲, ␮ = 0.237,
N = 0.763 共b兲, and ␮ = 0.0196, N = 0.935 共c兲.
Panel 共d兲 is a blow-up of 共b兲 in the region of
150⬍ x ⬍ 190.

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FIG. 3. The chemical potential vs norm for gap-soliton families in three lowest finite band gaps 共a,b,c兲 of the 1D model with ␧ = 1. The
dashed curves are added as guides to the eye.

␾共x,t兲 = e−i␮t␹共x兲, 共3兲 Families of the GS solutions are characterized by the de-
pendence of ␮ on the soliton’s norm, N = 兰−⬁ ␹ 共x兲dx. For GS
+⬁ 2

families in the three lowest finite band gaps, whose examples

are covered by vertical segments at fixed values of OL
were displayed in Fig. 2, the N共␮兲 dependences are plotted in
strength ␧. In Fig. 1共a兲, one can discern four finite gaps 共dis-
Fig. 3. Naturally, N → 0 corresponds to ␮ approaching the
continuities between the bands, the lowest gap being ex- border of the quasi-Bloch band located under the respective
tremely narrow兲 and the underlying semi-infinite gap 共one gap.
extending to ␮ → −⬁兲. A result of the standard perturbation The numerically found solution branches do not cover the
theory is that these gaps start 共at ␧ = 0兲 at ␮ = ␲2 / 2 and range of ␮ corresponding to the entire band gap in each case.
␮ = 共␲q兲2 / 2. Figure 1共b兲 details the results in a narrower However, the effective termination of the branches inside the
range of ␮ but for an extended interval of ␧. One can ob- gaps seems to be a numerical problem, rather than a true
serve, in particular, that the lowest band in Fig. 1共a兲 splits feature of the model: it is difficult to secure convergence of
into two bands at ␧ = 0.8 and three bands at ␧ = 1 in Fig. 1共b兲. the numerical solutions for very large values of N.
With the increase of ␧, additional band gaps open up on a Stability analysis of the GSs was performed by means of
finer scale 共accurate computation of the spectrum becomes direct simulations, using the split-step method with 4096
rather difficult for ␧ ⬎ 2兲. Fourier modes. This way, it was concluded that all the three
As the effective mass is negative near the top of families displayed in Fig. 3 are completely stable.
共quasi-兲Bloch bands, GSs are expected to exist in adjacent GSs obtained above can be readily set in motion in the
lower parts of the gaps. Assuming real ␹共x兲 and varying same way as was done in Ref. 关27兴, i.e., multiplying a sta-
␮ in Eq. 共3兲, we numerically searched for a family of tionary solution by exp共ikx兲, with k not too large. Examples
solutions of the corresponding stationary version of Eq. 共1兲, are displayed in Fig. 4, where the “shove factor” and the
␹⬙ / 2 = ␹3 + U共x兲␹ − ␮␹, satisfying boundary conditions resulting average velocity of the moving soliton are k = 0.03,
␹共x = L / 2兲 = A, ␹⬘共x = L / 2兲 = ␹共x = L兲 = 0. Figure 2 displays v = −0.104 共a兲; k = 0.015, v = −0.117 共b兲; and k = 0.005,
three typical examples of GSs with equal amplitudes, v = −0.03 共c兲. Accordingly, the effective GS mass M ⬅ k / v is
␹共x = L / 2兲 = A = 0.3, that were found in three finite band gaps −0.288 共a兲, −0.128 共b兲, and −0.17 共c兲. The mass is negative,
for ␧ = 1 at values of ␮ marked by crosses in Fig. 1共b兲 关as as it should be for GSs 关27兴. On the other hand, moving
usual 共in the GPE with self-repulsion兲, no solitons are found solitons feature conspicuous radiation losses if they are set in
in the semi-infinite gap兴. A blow-up of the GS in Fig. 2共b兲 is motion by shove k exceeding a certain critical value, which
shown in Fig. 2共d兲, to demonstrate that, while the envelope is kcr ⬇ 0.05, 0.017, and 0.008 for the solitons from Figs.
of the solution is that of a solitary wave, the carrier wave 2共a兲–2共c兲, respectively, although the transition to the lossy
function is nearly quasiperiodic. motion regime is not very sharp.

FIG. 4. Examples of stable moving gap solitons in three finite band gaps 共a,b,c兲 of the 1D model.

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HIDETSUGU SAKAGUCHI AND BORIS A. MALOMED PHYSICAL REVIEW E 74, 026601 共2006兲

FIG. 7. 共a兲 Evolution of 兩␾兩 for an unstable gap soliton labeled

by 3 in Fig. 6共a兲. 共b兲 Evolution of 兩␾共L / 2兲兩 in the gap soliton.

FIG. 5. Head-on collision of two gap solitons, which are ob- terms of Ref. 关27兴, as they are generated by the GPE with a
tained from ones displayed in Fig. 2共a兲 by shoving them with relatively small strength 共␧兲 of the pinning potential. For
exp共±ikx兲, k = 0.02. larger ␧, bands between the gaps in the model’s spectrum
become nearly invisible, and the character of GS solutions
Following the approach based on the separation of rapidly drastically alters, with a large variety of tightly bound soli-
and slowly varying functions, one can approximate GS solu- tons found in the broad band gaps. Figure 6共a兲 displays four
tions by different kinds of the solutions 共for ␧ = 5兲 with equal ampli-
tudes, 兩␾共x = L / 2兲兩 = 2. In the first potential well around the
␾共x,t兲 = e−i␮t␸␮共x兲⌽共x,t兲, 共4兲 central point, all the profiles are almost identical, but the next
where ␸␮共x兲 is a linear quasi-Bloch function pertaining to pair of peaks appears at markedly different positions, and
may even have different signs. The form of the quasiperiodic
chemical potential ␮, and ⌽共x , t兲 is a slowly varying ampli-
potential U共x兲 is displayed below the GSs. Naturally, local
tude, for which an averaged NLS equation can be derived as
maxima of 兩␾兩 are found near minima of the potential. In
in Ref. 关27兴,
fact, more GS species can be found, in addition to the four
⳵⌽ 1 ⳵ 2⌽ ones shown here. Figure 6共b兲 quantifies four GS families,
i =− + G兩⌽兩2⌽. 共5兲 whose representatives are displayed in Fig. 6共a兲, in terms of
⳵t 2M ⳵x2
the ␮共N兲 dependence. The curves overlap at sufficiently
Here M ⬍ 0 is the above-mentioned effective mass, and small N, as in this limit each GS species virtually reduces to
G ⬎ 0 is an effective nonlinearity coefficient, which can be the single peak in the first potential well; however, the spe-
found by matching obvious soliton solutions to Eq. 共5兲, cies become very different at larger N, corresponding to the
⌽ = A sech共A冑−MGx兲 共A is an arbitrary amplitude兲, to the different GS shapes displayed in Fig. 6共a兲. For example,
envelope of the numerically found GSs. As a result, for the branch 4 starts to deviate from the other three ones near
three quiescent GSs displayed in Fig. 2 we find G = 0.68 共a兲, N = 0.5, when negative peaks begin to develop near x = 16 and
0.41 共b兲, and 0.13 共c兲. The availability of stable moving soli- 24.
tons suggests to consider collisions between them. As pre- Direct simulations demonstrate that the GS species la-
dicted by the averaged equation 共5兲, moving GSs emerge beled as 1, 2, and 4 in Fig. 6 are stable, while the family
unscathed from collisions; see an example in Fig. 5. labeled by 3 is unstable. Further, Fig. 7共a兲 displays a typical
The GSs displayed above are loosely bound solutions, in example of the evolution of an unstable GS. The instability
breaks the soliton’s symmetry and makes it a breather. Al-
though the breathers’s amplitude features complex oscilla-
tions, see Fig. 7共b兲, the breather does not decay, maintaining
its localized shape.

FIG. 6. 共a兲 Four types of stationary gap solitons found in the 1D

model with a strong optical lattice, ␧ = 5. The chemical potential ␮
and the norm N are, respectively, ␮ = −1.261, N = 1.853 共1兲, FIG. 8. Contour plots of 兩␾共x , y兲兩 for three examples of stable
␮ = −1.088, N = 1.266 共2兲, ␮ = −1.214, N = 1.697 共3兲, and ␮ = gap solitons in the 2D model with a strong Penrose-tiling optical
−1.2415, N = 2.854 共4兲. The form of the potential is shown in the lattice, ␧ = 5. The solitons are generated by initial conditions 共8兲,
bottom. Panel 共b兲 displays curves ␮共N兲 for the four species of the with, respectively, ␮ = −1.56 共a兲, ␮ = −3.33 共b兲, and ␮ = −0.41 共c兲.
gap solitons. Solution branches shown in 共b兲 terminate at points The norms of the three solitons are Na = 5.61, Nb = 3.93, and
where the solitons quickly develop the tail structures. Nc = 10.2.

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FIG. 9. Time dependence of the peak amplitudes of three local- FIG. 10. Chemical potential ␮ of two-dimensional fundamental
ized solutions shown in Fig. 8. solitons vs N, in the model with ␧ = 5. Rhombuses, crosses, and
squares represent soliton families generated, respectively, by three
III. TWO-DIMENSIONAL SOLITONS initial configurations in Eq. 共8兲 共typical examples of solitons be-
longing to the three families are given in Fig. 8兲.
In the normalized form, the 2D version of the GPE with
the PT 共Penrose tiling兲 potential is

冋兺 册
They were generated by initial conditions 关in Eq. 共7兲兴 with,
5
⳵␾ 1 respectively,
i = − ⵜ 2␾ + 兩 ␾ 兩 2␾ − ␧ cos共k共n兲 · r兲 ␾ , 共6兲

冦 冧 冉 冊
⳵t 2 n=1 exp关− 0.5共x2 + y 2兲兴 5
1 共n兲
with 兵k共n兲 共n兲
x , k y 其 = ␲兵cos关2␲共n − 1兲 / 5兴 , sin关2␲共n − 1兲 / 5兴其. GS
⌽0共x,y兲 = 1.5 exp关− 0.15共x + y 兲兴
2 2
兺 cos
n=1 2
k ·r ,
solutions in two dimensions were constructed by means of a 2 exp关− 0.05共x2 + y 2兲兴
combination of the known method of the integration in 共8兲
imaginary-time 关28兴 and subsequent real-time simulations.
To this end, a substitution was first made, ␾共x , y , t兲 the last multiplier being a half-harmonic counterpart of the
= e−i␮t⌽共x , y , −i␶兲, which transforms Eq. 共6兲 into a nonlinear PT potential in Eq. 共6兲, which is expected to parametrically
diffusion equation, couple to the potential.

冋兺 册
Figure 8 displays contour plots of 兩␾共x , y兲兩 in final states
5
⳵⌽ 1 2 produced by the numerical integration. To illustrate the
= ⵜ ⌽ + 共␮0 − ⌽2兲⌽ + ␧ cos共k共n兲 · r兲 ⌽. 共7兲 proximity of the solutions to truly stationary ones, and
⳵␶ 2 n=1
their stability, in Fig. 9 we show the asymptotic time depen-
This equation was solved numerically by dint of the split- dence of the peak amplitudes of the three solitons at
step Fourier method with 256⫻ 256 modes, in a domain of 共x , y兲 = 共L / 2 , L / 2兲 = 共25, 25兲. These gap solitons are found in
the size of 50⫻ 50. It was observed that the norm of the a strong optical lattice with ␧ = 5, for which bands separating
solution originally decreased, and then began to increase. the spectral gaps are extremely narrow.
The imaginary-time integration was switched back into Systematic results for fundamental 2D solitons were
simulation of the GPE in real time when the norm attained its generated by using initial conditions 共8兲 and varying the
minimum. respective chemical potential ␮. Figure 10 shows ␮共N兲 de-
First, we present typical examples of stable fundamental pendences for three families of thus generated GSs, where
共zero-vorticity兲 2D solitons, for ␮ = −1.56, −3.33, and −0.41. N = 兰兰兩␾共x , y兲兩2dxdy is the usual 2D norm. At relatively small

FIG. 11. A typical example of a stable vortical gap soliton with S = −1, found in the model with ␧ = 5. 共a兲 and 共b兲 Contour plots of 兩␾共x , y兲兩
and Re ␾共x , y兲, the latter shown only in the region with Re ␾共x , y兲 ⬎ 0. 共c兲 A sequence of central cross sections, 兩␾共x = L / 2 , y兲兩, taken at
different moments of time, which demonstrate the vanishing of the field at the center, as must be in the vortex, and its stability. The norm
and chemical potential of the solution are N = 9.69 and ␮ = −3.

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HIDETSUGU SAKAGUCHI AND BORIS A. MALOMED PHYSICAL REVIEW E 74, 026601 共2006兲

values of N, the dependences completely overlap, and they supported by a weak lattice were found in three gaps; they
slightly split up at very large N, corresponding to ␮ → −0. are mobile, and collide elastically. The strong 1D lattice sup-
They are somewhat 共but not quite兲 similar to plots for tightly ports a variety of species of tightly bound solitons. Most of
bound GSs in the 1D model, cf. Fig. 6共b兲. There are no them are stable, unstable ones giving rise to robust asymmet-
loosely bound stable GSs in two dimensions, because the ric breathers. 2D gap solitons, both fundamental and vortical
averaged two-dimensional NLS equation 关a 2D counterpart ones, are stable too, and they may only have a tightly bound
of Eq. 共5兲兴 does not have stable soliton solutions. shape. The predicted solitons can be created by means of
Lastly, stable 2D solitons with embedded vorticity S can available experimental techniques in a Bose-Einstein con-
be found too. They were generated by adding a factor, densate with a positive scattering length, loading it into an
共x2 + y 2兲兩S兩/2 exp共iS␪兲, which distinguishes vortex states in optical lattice induced by a superposition of several laser
uniform media, to initial Ansätze 共8兲. A typical example of beams.
such a stable vortical gap soliton is presented in Fig. 11.
While the field pattern of 兩␾共x , y兲兩, displayed in Fig. 11共a兲, is ACKNOWLEDGMENTS
stationary, Fig. 11共b兲 is actually a snapshot, as the pattern of
Re ␾共x , y兲 shown in this panel rotates clockwise, at the an- B.A.M. appreciates the hospitality of the Department of
gular velocity of ␻ = ␮ / S = −3. Applied Science for Electronics and Materials at the Inter-
disciplinary Graduate School of Engineering Sciences, Ky-
IV. CONCLUSION
ushu University 共Fukuoka, Japan兲. This work was supported,
in part, by a Grant-in-Aid for Scientific Research No.
We have constructed families of 1D and 2D gap solitons 17540358 from the Ministry of Education, Culture, Sports,
in the model combining the self-defocusing nonlinearity and Science and Technology of Japan, and the Israel Science
quasiperiodic lattice potentials. Loosely bound 1D solitons Foundation through a Center-Excellence Grant No. 8006/03.

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