Localized modes in mini-gaps opened by periodically modulated intersite coupling in two-dimensional nonlinear lattices
Goran Gligorić, Aleksandra Maluckov, Ljupčo Hadžievski, Boris A. Malomed
aa r X i v : . [ n li n . PS ] M a y Localized modes in mini-gaps
Localized modes in mini-gaps opened by periodically modulated intersite coupling intwo-dimensional nonlinear lattices
Goran Gligori´c, a) Aleksandra Maluckov, Ljupˇco Hadˇzievski, and Boris A. Malomed (Dated: 21 May 2014)
Spatially periodic modulation of the intersite coupling in two-dimensional (2D) non-linear lattices modifies the eigenvalue spectrum by opening mini-gaps in it. Thiswork aims to build stable localized modes in the new bandgaps. Numerical analysisshows that single-peak and composite two- and four-peak discrete static solitons andbreathers emerge as such modes in certain parameter areas inside the mini-gaps of the2D superlattice induced by the periodic modulation of the intersite coupling alongboth directions.The single-peak solitons and four-peak discrete solitons are stablein a part of their existence domain, while unstable stationary states (in particular,two-soliton complexes) may readily transform into robust localized breathers.PACS numbers: 03.75.Lm; 05.45.YvKeywords: inter-site modulation, mini-gap solitons a) Electronic mail: [email protected]
Nonlinear lattices offer a possibility to create a vast variety of self-trapped (i.e.,spontaneously localized) discrete wave packets, alias discrete solitons. They aresupported by the stable balance between the onsite nonlinearity and discretediffraction in the lattice. In optics, discrete solitons have been observed in one-and two-dimensional (1D and 2D) arrays of nonlinear waveguides. In recentyears, such photonic lattices have been implemented as permanent structures,or optically induced as virtual ones, using various materials, including those withcubic, quadratic, photorefractive, and liquid-crystal nonlinearities. In these sys-tems, discrete solitary modes (fundamental and multi-peak solitons, 2D vortexsolitons, etc.) are observed under both the self-focusing (in-phase states) and de-focusing nonlinearity. In the former case, the discrete solitons have an in-phasestructure (in particular, the fundamental single-peak solitons are represented byreal positive solutions), while in the latter case solitons exist in the staggered form, with alternating signs of the discrete field at adjacent sites. Discrete soli-tons are also known in many other fields, such as Bose-Einstein condensates(BEC), electric transmission lines, solid-state lattice media, polymer molecules,etc.In this work, we address the formation and ensuing dynamics of localizedstructures in 2D photonic superlattices , induced by a relatively long-wave peri-odic modulation of the intersite coupling imposed on the underlying lattice withthe onsite cubic nonlinearity. In addition to photonics, similar BEC-trappingsettings can be built in the form of modulated optical lattices, by means of asuperposition of laser beams illuminating the condensate. We demonstrate thatthe periodic modulation of the intersite coupling opens narrow gaps in the linearspectrum of the 2D superlattices, and thus enables the creation of new solitarymodes ( gap solitons ) in these mini-gaps . Some of these modes are robust (notonly static ones, but also periodically oscillating breathers ), therefore they canbe experimentally created in the photonic and matter-wave (BEC) settings.
I. INTRODUCTION
Discrete solitons represent self-trapped states in nonlinear lattice systems. They resultfrom the interplay between the lattice diffraction and material nonlinearity. In optics, thesestates have been experimentally observed in both one- and two-dimensional (1D and 2D)nonlinear waveguiding arrays. Such photonic lattices have been built as permanent struc-tures, or induced as virtual ones, in a variety of optical media, including those with cubic,quadratic, photorefractive (saturable), and liquid-crystal nonlinearities . Similar spa-tially periodic trapping structures for quasi-discrete states of matter waves, based on opticallattices (induced by the interference of laser beams illuminating the Bose-Einstein conden-sate, BEC), have been created too . In addition, discrete solitons are known in chains ofmicromechanical oscillators , liquid crystals , biological macromolecules , and in othersettings.Discrete fundamental and vortical solitons in 2D nonlinear waveguide arrays werefirst observed in biased photorefractive crystals. Properties of fundamental discrete solitonswere studied in detail theoretically too . In these systems, solitons exist in unstaggered(i.e., in-phase) and staggered (i.e., with phase shift π between adjacent lattice sites) forms,under the self-focusing (SF) and self-defocusing (SDF) onsite nonlinearity, respectively. Dis-crete solitary vortices were also studied in detail by means of numerical methods . Inparticular, attention has been drawn to the 2D band structure associated with photonics lat-tices and respective aspects of the stability of 2D discrete solitons, see, e.g., Ref. . Specificfeatures of lattice solitons in BEC models were addressed in Refs. .Discrete localized modes were also studied in inhomogeneous 1D lattices, subject to aquasiperiodic spatial modulation of the intersite coupling constant , as well as with an in-homogeneous onsite nonlinearity . The modulation of the lattice coupling constant merelyimplies a varying spacing between the sites, as the coupling constant depends on it expo-nentially. An interesting finding is that the strength of the onsite SDF nonlinearity, whichgrows from the center to periphery of the 1D lattice at any rate faster than the distance, | n | ,supports solitons of the unstaggered type, which are impossible in the uniform lattice withthe SDF sign of the nonlinearity . Equivalently, solitons of the staggered type are supportedby the growing onsite SF nonlinearity, which is not possible either in the uniform lattice. Itwas demonstrated too that unstaggered solitons exist in the lattice with homogeneous onsite3ocalized modes in mini-gapsSDF nonlinearity, if the coupling constant decays fast enough at | n | → ∞ .In this paper, we aim to extent the study of discrete solitons to 2D inhomogeneous lattices,namely, to those with periodic modulation of the intersite coupling. These settings may beconsidered as superlattices , which, in the general case, are defined as structured createdby imposing relatively long-wave periodic spatial modulations onto the underlying lattice .The model, based on the corresponding discrete nonlinear Schr¨odinger (DNLS) equationswith the SF onsite nonlinearity, is introduced in Section II. First, we study linear propertiesof the modulated 2D lattices, and demonstrate opening of mini-gaps in the correspondinglinear spectrum, where self-trapping of new types of discrete solitons may be expected. Forthe sake of the completeness, in subsection III.A we briefly present results of the study ofself-trapped modes (fundamental and vortex solitons) residing in the semi-infinite spectralgap in the superlattice, and compare them to the corresponding results in uniform lattices.The main topic of this work, which is presented in subsection III.B, is the creation of stablediscrete solitons in the mini-gaps. We find families of fundamental solitons in mini-gaps, aswell as two- and four-soliton complexes. While the fundamental solitons are stable, numericalsimulations demonstrate that the complexes evolve into localized breathing structures. Thepaper is concluded by Section IV. II. THE MODELA. Basic equations
The 2D discrete model is based on the following DNLS equation, with the cubic onsiteSF nonlinearity, for complex field amplitudes ψ m,n : i dψ m,n dz + C m,n ( ψ m +1 ,n + ψ m − ,n ) + K m,n ( ψ m,n +1 + ψ m,n − ) + γ | ψ m,n | ψ m,n = 0 , (1)where the horizontal and vertical coupling constants are modulated, respectively, along thehorizontal and vertical directions, as shown in Fig. 1: C m,n = C [1 + ∆ cos( Q a m )] ≡ C m K m,n = C [1 + ∆ cos( Q b n )] ≡ K n . (2)This setting can be easily implemented in photonic lattices by properly selecting distancesbetween the constituent waveguides, as well as for the BEC loaded into deep optical lattices,4ocalized modes in mini-gapsshaped as shown in Fig. 1. In the former case, evolution variable z is the propagation dis-tance along individual waveguides, while in the matter-wave (BEC) realization, z is replacedby time t . FIG. 1. (a) The schematic plot of the lattice with corresponding to modulation pattern (2) with Q a = Q b = π , see also Eq. (8). (b) A fragment of the superlattice corresponding to Q a = Q b = π/ In addition, we also considered the 2D lattice with another modulation pattern, corre-sponding to the horizontal and vertical couplings periodically modulated along the verticaland horizontal directions, respectively: C m,n = C [1 + ∆ cos( Q a n )] K m,n = C [1 + ∆ cos( Q b m )] . (3)Unlike the model based on Eq. (1), the analysis has not revealed any dynamically stablelocalized structures in Eq. (3), except for the usual onsite solitons in the semi-infinitespectral gap, and it is not straightforward to implement the latter model experimentally.Therefore, we here focus on Eq. (2).The results are presented below for γ = +1 (which corresponds to the SF nonlinearity), C = 1, and ∆ = ∆ = 0 . Q a = Q b = π/ Q a = Q b , demonstrates that this case adequately represents the genericsituation. 5ocalized modes in mini-gapsStationary solutions to Eqs. (1) with real propagation constant − µ (or chemical potentialfor the BEC) are looked for as ψ m,n ( z ) = e − iµz U m,n , (4)where stationary discrete function U m,n obeys the following equation: µU m,n + C m ( U m +1 ,n + U m − ,n ) + K n ( U m,n +1 + U m,n − ) + | U m,n | U m,n = 0 , (5)with the coupling constants defined as per Eq. (2). The power of the discrete soliton isdefined as usual, P = X m,n | U m,n | . (6)Stationary equation (5) was solved by means of a numerical algorithm based on the mod-ified Powell minimization method . Stability of the so found discrete solitons was checked,in the framework of the linear stability analysis, by numerically solving the correspondingeigenvalue equation for modes of small perturbations. Finally, the evolution equation (1)was directly simulated by dint of the Runge-Kutta procedure of the sixth order, cf. Ref. .The simulations were used to verify the stability properties predicted by the linear analysis. B. The linear spectrum
The linearized version of Eq. (5) is µU m,n + C m ( U m +1 ,n + U m − ,n ) + K n ( U m,n +1 + U m,n − ) = 0 . (7)Its eigenvalue (EV) spectrum can be derived analytically for binary lattices , which arecharacterized by alternating values of the coupling constants ( Q a = Q b = π ), see Fig. 1(a): C m = C [1 + ( − m ∆ ] = 1 ± . ≡ C ,K m = C [1 + ( − n ∆ ] = 1 ± . ≡ K , (8)with the top and bottom signs corresponding, severally, to even and odd values of m or n .The corresponding solution for amplitudes in this case can be looked for as( a m,n , b m,n , c m,n , d m,n ) = ( A, B, C, D ) exp [ i ( κ a m + κ b n )] , (9)6ocalized modes in mini-gapswhere κ a , κ b are the Bloch wavenumbers in the m and n directions, while a, b, c, d pertain tofour sets of sites distinguished by the four different values of the coupling constants in Eq.(8). The substitution of ansatz (9) into Eq. (7) leads to the following eigenvalue equation:16 C C cos κ a − (cid:0) cos κ a (cid:1) (cid:2) ( C + C ) µ + 8 C C K K cos κ b (cid:3) + (cid:0) µ − K cos κ b (cid:1) ( µ − K cos κ b ) = 0 . (10)For κ a,b = π/
2, Eq. (10) degenerates to µ = 0, at which point the gaps get closed. In thissituation, we have found a few different families of nonstationary localized solutions, whichradiate due to coupling to linear lattice modes in the absence of the gap.In the general case, the eigenvalue problem was solved numerically. It has been foundthat, in addition to the semi-infinite gap, the spectrum of the superlattice contains newnarrow ( mini -) gaps, in which the linear Bloch waves do not exist, while their nonlinearcounterparts are modulationally unstable, thus opening the way to create localized struc-tures (discrete gap solitons) with the propagation constant falling into the mini-gaps , viathe interplay of the nonlinearity, discreteness and modulated intersite coupling. Our mainobjective here is to demonstrate that some of such gap solitons are stable. They can beobserved experimentally , and may be used to control the light propagation in photonicstructures. It is relevant to mention that mini-gaps, and gap solitons existing in them,are known in continual nonlinear models of Bragg supergratings, i.e., gratings subject to along-wave modulation .As an illustration, the linear spectrum for the lattice with Q a = Q b = π/ | µ | = 4, which are placed symmetrically withrespect to µ = 0. Extensive numerical calculations, performed for lattices with differentratios between Q a and Q b , produce similar spectra. III. LOCALIZED MODES
It is known that 2D nonlinear lattices with the uniform intersite coupling give rise tofundamental bright discrete solitons of the unstaggered type, and different types of vorticesin the parameter region corresponding to the semi-infinite gap in the corresponding linearspectrum . In the case of the 2D lattices with the SDF onsite nonlinearity, bright solitonsare produced by the staggering transformation .7ocalized modes in mini-gaps
00 1 2 3 4x10 -4 m mode number FIG. 2. The linear spectrum for the lattice with Q a = Q b = π/ = ∆ = 0 .
5. Thespectrum features semi-infinite gaps at (approximately) | µ | > .
73, and mini-gaps around | µ | = 4(both presented by gray areas). Similar spectra, with slightly different positions and widths of thegaps, are obtained for other values of Q a , Q b . In comparison with the uniform lattice, the periodic modulation of the intersite couplingopens the mini-gaps, in which new discrete solitons are expected, as said above. We demon-strate below that discrete solitons residing in the semi-infinite gap of the modulated latticesare not significantly altered by the spatially periodic inhomogeneity, while completely novelspecies of staggered discrete solitons are found in the mini-gaps.
A. Soliton families in the semi-infinite gap
Three types of fundamental unstaggered solitons have been found in the semi-infinite gap( µ < − .
73 in Fig. 2): onsite, hybrid, and intersite ones, see Fig. 3. They feature dynamicalproperties similar to those of their counterparts in the 2D uniform lattice. In particular,8ocalized modes in mini-gapssolely the onsite family is stable, in almost the entire existence region, see Fig. 4. Themodulation of the lattice only slightly extends the area where the stable onsite modes arefound.Vortex solitons with topological charge S = 1 and S = 2 are formed too in the semi-infinite gap. In terms of their stability and dynamics, they are also similar to their counter-parts in the 2D uniform lattice. In particular, vortices with topological charges S = 1 and S = 2 are stable in certain parts of their existence region. The anisotropic intersitecoupling in the modulated 2D lattice affects the shape of the solitons, as illustrated in Fig.3 for the intersite and hybrid fundamental solitons. -2 0 2-202 n m -4 -2 0 2-4-202 n m -4 -2 0 2-4-202 n m -20 -15 -10 -503060 P m on-sitehybridinter-site (a) (b)(c) (d) FIG. 3. (Color online) Amplitude profiles of the 2D solitons belonging to the semi-infinite gapin the superlattice created by periodic modulation (3) with Q a = Q b = π/
3: (a) onsite-centered,(b) inter-site-centered, and (c) hybrid fundamental solitons. Plot (d) shows the dependence of thesoliton’s norm P vs. µ for the fundamental modes (solid line - onsite, dashed line - hybrid, dottedline - inter-site); the norm is defined as per Eq. (6). -20 -15 -10 -55101520 P FIG. 4. (Color online) The P ( µ ) dependence for 2D onsite solitons belonging to the semi-infinitegap: black solid and red dashed lines correspond to the uniform and periodically modulated ( Q a = Q b = π/
3) lattices, respectively. The stability region for the uniform and modulated lattices arelocated, severally, on the left of the dotted black and red vertical lines.
B. Soliton families in the mini-gaps
1. Fundamental solitons
Several different single-soliton families have been found in the mini-gaps. We here considerthe single one, which is stable in a part of its existence region, see Fig. 5(a), while all theother families are completely unstable. The P ( µ ) dependence for the soliton family is shownin Fig. 5(b). By approaching both edges of the mini-gap the single soliton families disappearin the sense that corresponding localized patterns cannot be created. In the lower bound P vanishes, while in the upper grows. In both cases background is characterized by highlyirregular amplitudes with corresponding magnitude (small and high, respectively) withoutclearly distinguishing localized structure. It is found that, except for the light-gray area,3 . < µ < . FIG. 5. (Color online) Solitons in the mini-gap opened around µ = 4, cf. Fig. 3. (a) A stablesoliton found for µ = 4 .
05 [this value of µ is designated by the right green vertical dashed line inpanel (b)]. (b) The solitons’s norm P vs. µ . The gray rectangle designates the stability region ofthe soliton’s type presented in plot (a). Plots (c) and (d) show the amplitude at the central site vs.time, produced by direct simulations of Eq. (1) at µ = 3 .
54 [a strongly unstable soliton, designatedby the left green vertical dashed line in panel (b)], and µ = 4 .
05 [the stable soliton displayed in(a)].
2. Solitons complexes
In addition to dynamically stable single-soliton gap modes, we have found different boundstates of solitons in the minigap. First, two-soliton complexes are built of two identicalsingle-soliton constituents, see Fig. 6. It is found that the two-soliton complex keeps itscompactness and the staggered structure in the course of perturbed evolution, which givesrise to regular amplitude oscillations, as seen in 6(c). In direct simulations, the two-soliton11ocalized modes in mini-gapscomplex is actually found to be more robust than predicted by the linear-stability anal-ysis, which demonstrate the presence of pure real EVs for small perturbations, i.e., weakinstability of such complexes.
FIG. 6. (Color online) Dynamics of two-soliton complexes in the mini-gap opened around µ = 4.(a) The profile of the two-peak complex ( µ = 4). (b) The norm of the complex, P , vs. µ (c) Theamplitudes of the constituent solitons vs. time, in the complex designated by the vertical line in(b). On the other hand, four-soliton complexes, which are built of four identical in-phaseindividual solitons (with zero phase shifts between them), see Fig. 7, feature stabilityproperties similar to those of the constituent solitons, as shown in Fig. 7(b). In particular,the instability of the four-soliton complex is accounted for by purely real EVs, when they arepresent, and the complex is stable in the absence of such eigenvalues, which is corroboratedby direct simulations. Thus, the in-phase four-soliton complexes are essentially more stablethan their two-soliton counterparts. On the other hand, out-of-phase composites were alwaysfound to be unstable, independently on the number of constituent solitons. Note that bothmulti-soliton families cannot be formed in the neighborhood of the mini-gap edges. Thecorresponding soliton branches are changed by solution branches corresponding to more orless irregular background similarly to the case of fundamental solitons.In the case of the SDF nonlinearity, single-soliton modes and their two- and four-solitonbound states can be generated by the staggering transformation from their unstaggeredcounterparts found for the SF nonlinearity in the mini-gap opened around µ = −
4, seeFig. 2. Naturally, these staggered modes feature the same stability and dynamics as theircounterparts which were considered above. 12ocalized modes in mini-gaps
FIG. 7. (Color online) Four-soliton complexes in the mini-gap. (a) The profile of the stablecomplex at µ = 4 .
05. (b) The norm, P , vs. µ , for the family of the four-soliton complexes. Thegray rectangle denotes the stability region for solitons of the type presented in panel (a). (c)Amplitudes of the constituent solitons vs. time, for the stable complex designated by the verticalline in (b), in the course of its perturbed evolution. C. Other solutions
Direct simulations clearly demonstrate that all the modes observed in the mini-gapsare immobile in the 2D lattice: in direct simulations, the application of a kick to stablemodes, in any direction with respect to the underlying superlattice, causes oscillations ofthe kicked solitons, but not their progressive motion (not shown here in detail). In fact,such simulations, although they fail to produce motile discrete solitons, corroborate theirstability against positional perturbations.The lack of the mobility of the discrete solitons in the present model is not surprising,taking into regard the general property of immobility of 2D discrete solitons supported by thecubic nonlinearity , which is, in turn, explained by the vulnerability of the continuum-limitcounterparts of such solitons to the collapse. As a result, a soliton will be compressed bythe (quasi-) collapse until it will become strongly localized, on few lattice sites, i.e., stronglypinned to the lattice, which provides for its stabilization but prevents it from being mobile.It is known that the mobility of discrete solitons may be enhanced by special modes of“management”, i.e., by the application of time-periodic [or z -periodic, in terms of Eq. (1)]modulation of the local nonlinearity (at least, in 1D settings) . The consideration of suchmechanisms may be a subject for a separate work, being beyond the scope of the presentone. 13ocalized modes in mini-gapsLastly, in the superlattice with the spatially periodic modulation of the intersite couplingdefined as per Eq. (3), new gaps in the linear spectrum open too. In those gaps, differentstationary localized solutions can be found. The linear-stability analysis predicts that allthese states are subject to an oscillatory instability, while in direct simulations some of themmay seem as robust breathers, which is explained by the saturation of the instability withthe increase of the oscillation amplitude. IV. CONCLUSION
We have demonstrated that periodic modulation of the intersite coupling opens narrowgaps in the linear spectrum of the 2D square-shaped lattices, offering the possibility tocreate new species of discrete solitons in these mini-gaps, if the onsite cubic nonlinearityacts in the system. A part of the family of fundamental solitons and the respective four-soliton complexes are dynamically stable. Some other modes, which are unstable, developinto robust localized breathers. These modes can be experimentally created in arrays ofnonlinear optical waveguides and in BEC trapped in a deep optical lattice, be means ofcurrently available techniques - . It may be interesting to extend the analysis to 2D latticeswith different geometries, such as triangular, honeycomb, and quasi-periodic. ACKNOWLEDGMENTS
G.G., A.M., and Lj.H. acknowledge support from the Ministry of Education, Science andTechnological Development of the Republic of Serbia (Project III45010).
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