Compactons in Nonlinear Schrödinger Lattices with Strong Nonlinearity Management
aa r X i v : . [ n li n . PS ] N ov Compactons in Nonlinear Schr¨odinger Lattices with Strong Nonlinearity Management
F.Kh. Abdullaev, P.G. Kevrekidis, and M. Salerno CFTC, Universidade de Lisboa, Av. Prof.Gama Pinto 2, Lisboa 1649-003, Portugal Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003 Dipartimento di Fisica “E.R. Caianiello”, CNISM and INFN - Gruppo Collegato di Salerno,Universit`a di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy
The existence of compactons in the discrete nonlinear Schr¨odinger equation in the presence of fastperiodic time modulations of the nonlinearity is demonstrated. In the averaged DNLS equation theresulting effective inter-well tunneling depends on modulation parameters and on the field amplitude.This introduces nonlinear dispersion in the system and can lead to a prototypical realization ofsingle- or multi-site stable discrete compactons in nonlinear optical waveguide and BEC arrays.These structures can dynamically arise out of Gaussian or compactly supported initial data.
PACS numbers: 42.65.-k, 42.81.Dp, 03.75.Lm
Introduction . One of the most remarkable phenomenaoccurring in nonlinear lattices is the existence of discretebreathers which arise from the interplay between discrete-ness, dispersion and nonlinearity [1]. These excitationsare quite generic in nonlinear lattices with usual (e.g. lin-ear) dispersion and have typical spatial profiles with ex-ponential tails. In presence of nonlinear dispersion theseexcitations (as well as their continuous counterparts) mayacquire spatial profiles with compact support and for thisreason they are known as compactons [2]. Unlike othernonlinear excitations, compactons (having no tails) can-not interact with each other until they are in contact,this being an attractive feature for potential applications.Similarly to discrete breathers, compactons are intrinsi-cally localized and robust excitations. The lack of expo-nential tails is a consequence of the nonlinear dispersiveinteractions which permit the vanishing of the intersitetunneling at compacton edges. The difficulty of imple-menting this condition in physical contexts has restricteduntil now investigations mainly to the mathematical side.The development of management techniques for solitoncontrol, however, can rapidly change the situation.Periodic management of parameters of nonlinear sys-tems has been shown to be an effective technique for thegeneration of solitons with new types of properties [3].Examples of the management technique in continuoussystems are the dispersion management of solitons in op-tical fibers which allows to improve communication ca-pacities [4], and the nonlinearity management of 2D and3D Bose-Einstein condensates (BEC) or optically lay-ered media which provides partial stabilization againstcollapse in the case of attractive interatomic interac-tions [5]. In discrete systems the diffraction managementtechnique was used to generate spatial discrete solitonswith novel properties [6, 7] which have been observedin experiments [7]. The resonant spreading and steeringof discrete solitons in arrays of waveguides, induced bynonlinearity management was also investigated [8]. Todate, the nonlinear management technique for nonlin-ear lattices has been considered only in the limit of weakmodulations of the nonlinearity [9, 10]. The inter-welltunneling suppression has been discussed in [11] for the Bose-Hubbard chain with time periodic ramp potentialand in [12] for a two-sites Bose-Hubbard model with mod-ulated in time interactions. In both cases the tunnelingsuppression was uniform in the system and no appar-ent link with compacton formation was established. Thephenomenon has also been recently observed in experi-ments of light propagation in waveguide arrays [13] andin BEC’s in strongly driven optical lattices [14].The aim of the present Letter is to demonstrate theexistence of stable compacton excitations in the dis-crete nonlinear Schr¨odinger (DNLS) system subjected to strong nonlinearity management (SNLM), e.g. to fast pe-riodic time variations of the nonlinearity. To that effect,we use an averaged DNLS Hamiltonian system to showthat in the SNLM limit the inter-well tunneling can betotally suppressed for field amplitudes matching zeros ofthe Bessel function, introducing effective nonlinear dis-persion which leads to compacton formation. We showthat these compact structures not only exist in singleand multi-site realizations but they generically are struc-turally and dynamically stable and can be generated fromgeneral classes of initial conditions. These results shouldenable the observation of discrete compactons in BECand in nonlinear optical systems, both being describedby the discrete NLS equation.
Theory.
Consider the following lattice Hamiltonian H = − X n { κ ( u n u ∗ n +1 + u n +1 u ∗ n ) + 12 ( γ + γ ( t )) | u n | } , (1)with the coupling constant κ quantifying the tunnelingbetween adjacent sites (wells), γ denoting the onsiteconstant nonlinearity and γ ( t ) representing the time-dependent modulation. In the following we assume astrong management case with γ ( t ) being a periodic, e.g. γ ( t ) = γ ( t + T ), and rapidly varying function. As aprototypical example, we use γ ( t ) = γ ε cos(Ω τ ), with γ ∼ O (1), ε ≪ τ = t/ε denoting the fast time vari-able and T = 2 π/ Ω the period. The dynamical systemassociated with (1) is the well known DNLS equation [15] i ˙ u n + κ ( u n +1 + u n − ) + ( γ + γ ( t )) | u n | u n = 0 , (2)which serves, under suitable conditions [16], as a modelfor the dynamics of BEC in optical lattices subjectedto SNLM (through varying the interatomic scatteringlength by external time-dependent magnetic fields via aFeshbach resonance), as well as for light propagation inoptical waveguide arrays (here the evolution variable isthe propagation distance and the SNLM consists of peri-odic space variations of the Kerr nonlinearity).The existence of compacton solutions can be inferredfrom the fact that the averaged DNLS Hamiltonian (av-eraged with respect to the fast time τ ), coincides withthe original time independent Hamiltonian except for arescaling of the coupling constant which depends on theBessel function of the field amplitude. To show this, itis convenient to perform the transformation [17] u n ( t ) = v n ( t ) e i Γ | v n ( t ) | with Γ = ǫ R t dt γ ( τ ) = γ Ω − sin(Ω τ ),which allows to rewrite Eq. (2) as i ˙ v n = Γ v n ( | v n | ) t − κX − γ | v n | v n , (3)with X = v n +1 e i Γ θ + + v n − e i Γ θ − and θ ± = | v n ± | −| v n | .On the other hand, ( i | v n | ) t = i ( ˙ v n v ∗ n + v n ˙ v ∗ n ) = iκ ( v ∗ n X − v n X ∗ ), with the star denoting the complex conjugation.Substituting this expression into Eq. (3) and averagingthe resulting equation over the period T of the rapidmodulation, we obtain i ˙ v n = iκ | v n | h Γ X i− iκv n h Γ X ∗ i− κ h X i− γ | v n | v n , (4)with h·i ≡ T R T ( · ) dτ denoting the fast time average.The averaged terms in Eq. (4) can be calculated bymeans of the elementary integrals h e ± i Γ θ ± i = αJ ( αθ ± ), h Γ e ± i Γ θ ± i = ± iαJ ( αθ ± ), with J i being Bessel functionsof order i = 0 , α = γ / Ω, thus giving i ˙ v n = − ακv n [( v n +1 v ∗ n + v ∗ n +1 v n ) J ( αθ + ) +( v n − v ∗ n + v ∗ n − v n ) J ( αθ − )] − κ [ v n +1 J ( αθ + ) + v n − J ( αθ − )] − γ | v n | v n . (5)Note that parameters γ , Ω ∼
1, and the averaged equa-tion is valid for times t ≤ /ǫ . This modified DNLSequation can be written as i ˙ v n = δH av /δv ∗ n , with aver-aged Hamiltonian H av = − X n { κJ ( αθ + ) (cid:2) v n +1 v ∗ n + v ∗ n +1 v n (cid:3) + γ | v n | } . (6)A comparison with Eq. (1) gives the anticipated rescal-ing as κ → κJ ( αθ + ); a similar rescaling was recentlyreported also for a quantum Bose-Hubbard dimer withtime dependent onsite interaction [12].It is worth noting that while the appearance of theBessel function is intimately connected with harmonicmodulations, the existence of compacton solutions andthe lattice tunneling suppression is generic for peri-odic SNLM. Thus, for example, for a two-step mod-ulation of the form γ ( t ) = ( − i γ with i = 0 , i < τ < ( i +1)2 , we obtain for the first term −5 0 5012 u n n−1 0 1−4−2024 λ i λ r −5 0 5012 u n n−1 0 1−4−2024 λ r λ i −5 0 5−202 u n n−1 0 1−4−2024 λ i λ r −5 0 5012 u n n−1 0 1−505 λ i λ r FIG. 1: Typical examples for κ = 0 . α = 1 of compact local-ized mode solutions of Eq. (7) (top panels) and of the plane( λ r , λ i ) of their linearization eigenvalues λ = λ r + iλ i . 1st col-umn: on-site, 2nd column: inter-site, in-phase, 3rd column:inter-site, out-of-phase, 4th column: three-sites. Remarkably,all solutions are dynamically stable . in the averaged Hamiltonian (6) κ ( v ∗ n +1 v n e iγ θ + / + v ∗ n v n +1 e − iγ θ + / )sinc( γ θ + / x ) = sin( x ) /x ,thus, in this case the suppression of tunneling exists atzeros of the sinc function. We also remark that for small αθ + the series expansion of J yields the averaged Hamil-tonian of the DNLS equation obtained in [9] in the limitof weak management. Exact compactons and numerics . To demonstrate theexistence of exact stable compactons in the averagedsystem, we seek for stationary solutions of the form v n = A n e − iµt for which Eq. (5) becomes µA n + γ A n + κ ( A n +1 J ( αθ + ) + A n − J ( αθ − )) +2 ακA n [ A n +1 J ( αθ + ) + A n − J ( αθ − )] = 0 . (7)As is well known, discrete breathers can be numericallyconstructed with high precision using continuation pro-cedures from the anti-continuous limit. The applicationof this method to Eq. (7) gives, quite surprisingly, thatsuch modes cannot be continued past a critical point (of κ ≈ .
32 for − µ = γ = 1). The fact that the solu-tions cease to exist before reaching the limit of resonancewith the linear modes ( κ = − µ/
2) naturally raises thequestion of what type of modes may be present in thesystem for larger values of the coupling. In the followingwe show that in agreement with our theoretical predic-tion, the emerging excitations are genuine compactonse.g. they have vanishing tails (rather than fast doubleexponential decaying tails as in granular crystals [19]).To search for compactly supported solutions one needsto consider [18] the last site of vanishing amplitude, de-noted as n below. In the setting of Eq. (7), this directlyestablishes the condition J ( αA n +1 ) = 0 ⇒ A n +1 = 2 . /α (8) t n
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FIG. 2: Space-time evolution of a one-site (left column) anda two-sites (right column) compacton solution as obtainedfrom direct numerical integrations of Eq. (2). Top panels ineach case show the square modulus of the solution itself (largeamplitude colorbar), while bottom panels (small amplitudecolorbar) show the deviation from the exact solution of Eq.(7) taken as corresponding initial condition. which yields the solution (based on the first zero of theBessel function) for the “boundary” of the compactlysupported site. Then, for µ = − γ A n +1 , both the con-dition for compact support at n ±
1, and the equationfor n = n are satisfied. Hence Eq. (8) yields a single-sitediscrete compacton. Numerical linear stability analysisillustrates that this solution is generically stable (see Fig.1). The bottom panel’s eigenvalues are associated withperturbations growing as e λt . The absence of a positivereal part in λ (i.e. of any λ ’s in the right half plane) istantamount to linear stability. Similar results are foundfor two-site compactons, which are either in phase (2ndcolumn of Fig. 1) or out-of-phase (3rd column of Fig.1). The only thing that changes here is that in order tosatisfy the equation at the non-vanishing sites, one musthave µ = − κ − γ A n +1 , µ = κ − γ A n +1 , respectivelyfor the in-phase and out-of-phase two-site compactons(note from Fig. 1 that these solutions are both stable).With some additional effort, one can generalize theseconsiderations to an arbitrary number of sites. As a typ-ical example, a three-sites compacton with amplitudes( . . . , , A , A , A , , . . . ) will satisfy in addition to the“no tunneling condition” J ( αA ) = 0, the constraints: µA i + 2( i + 1) ακA i A − i J ( α ( A − i − A i )) + γ A i + κA − i J ( α ( A − i − A i )) = 0 , i = 0 , dynamically robust herein. This depar-ture from the standard DNLS model can be rationalizedby the fact that in the latter case the instability is me-diated by the intersite tunneling/coupling [15], which for n t
10 20 30 40−10010 n t
10 20 30 40−10010 123450.511.522.5 −5 0 5012 u n n −5 0 5012 u n n−5 0 5012 u n n−5 0 5−202 u n n FIG. 3: Left panel: dynamical evolution for κ = 1 of a per-turbed 3-sites compacton for ǫ = 0 . ǫ = 0 .
025 (bottom). Right panel: Ex-amples of stable large amplitude compact modes with one-,two- (in-phase and out-of-phase) and 3-sites emerging fromthe second zero of the Bessel function. our special compacton solutions vanishes, hence endow-ing the solutions herein with dynamical stability.The dynamical stability of the solutions of Fig. 1 withrespect to the original DNLS model in Eq. (2) has beeninvestigated in Fig. 2 for the one-site (left panels) andthe two-site, in-phase (right panels) modes (similar find-ings were obtained for other modes). The top panelsshow the space-time contour map of the solution mod-ulus, while the bottom panels illustrates the deviationfrom the initial condition. The structural stability ofthese compactons was ensured by adding a uniformly dis-tributed random perturbation of small amplitude to theoriginal solution. Both for the averaged equation (notshown here) and for the original system (see Fig. 2),the relevant perturbation stays bounded and never ex-ceeds 2% of the solution amplitude. The waveforms re-main remarkably localized in their compact shape (aftera transient stage of shedding off small amplitude “radia-tion”) and their tails never exceed an O( ǫ ) correction, astheoretically expected for timescales of O(1 /ǫ ). Noticethat for Eq. (2), γ ( t ) = 1 + ǫ cos( t/ǫ ), with ǫ = 0 . However, if one departs from the regime of validityof the averaging and from the SNLM limit, interestingdeviations from the above behavior (and stability) arise.An example of this is shown in the left panels of Fig. 3.In this case, the three-site solution was initialized in Eq.(2) with ǫ = 0 . ǫ = 0 .
025 inthe bottom one. In the latter, the above argued robust-ness of the averaged modes was observed. Yet, in theformer one, the apparent lack thereof was clearly due tothe use of an ǫ outside of the regime of applicability ofthe averaging approximation. Nevertheless, the result-ing evolution has two interesting by-products. Firstly, itconfirms the general preference of the system towards set-tling in compact modes, since the evolution asymptotes t n −2 | u n ( ) | n 12345 -4 0 4 8 12 1602468 t | u n ( ) | n FIG. 4: Left panel: Space-time evolution of a Gaussianwavepacket u n (0) = 1 . e − . n under Eq. (2) for κ = 0 . t = 200). Right panel: Three-site compacton at time t = 200 generated from uniform i.c. u i = 2 . , i = − , , κ = 0 .
5. The inset shows time evo-lution curves (from top to bottom) of amplitudes at 0 , , , to an essentially single-site solution. Secondly, the largeramplitude of this solution in comparison to those of Fig.1 led us to explore the possibility of compactly supportedmodes associated with higher zeros of the Bessel functionin the right panels of Fig. 3. Remarkably, such solutionsagain, not only exist but are stable in all the cases shownin the figure (numerical linear stability graphs are omit-ted). This indicates the existence of an infinite sequenceof such modes, connected with the zeros of the Besselstructure of the (averaged) tunneling.To address the robust emergence of such compact ex-citations, we used both few-sites uniform and even Gaus-sian type excitations. While in the former (experimen-tally realizable, see e.g. [20]) case, discrete compactonscan be expected, remarkably, in either case such exci-tations can result. A typical example is shown for aGaussian initial profile in the left panel of Fig. 4, which yields a single-site compact mode differing in amplitudeby more than two orders of magnitude between the cen-tral site and its nearest neighbor and showing no signswhatsoever of an exponential tail , even in a semilog plot.In the right panel of Fig. 4 a multi-site compacton gener-ated from uniform compactly supported data is depicted.We see that the amplitude reduces by five orders of mag-nitude 3-sites away from the central peak.Let us estimate the parameters for the experimentalobservation of such modes, e.g. for the case of Li con-densate in a deep optical lattice. The Feshbach reso-nance in Li occurs at the value of external magnetic field B = 720 G . By varying the magnetic field around thisvalue we can easily obtain variations of the scatteringlength a s around the order of the background scatter-ing length a s yielding γ /γ ∼
10. In the deep opticallattice with V > E R , where V is the depth of the lat-tice and E R = ~ k / m is the recoil energy, the Gross-Pitaevskii equation can be mapped into the DNLS equa-tion (2) [16]. Thus by changing periodically in time themagnetic field between these values with the frequencyΩ ∼ ω R , where ω R = E R / ~ , we can generate matterwave compactons. Conclusions . We predicted the existence of discrete com-pactons in the DNLSE with strong nonlinearity manage-ment. We found stable single and few-sites compactonsof odd and even parity. They are robust and can be gen-erated from different classes of initial conditions. Suchstructures may be observable in experiments on BECs indeep optical lattices with periodically varying scatteringlength and arrays of nonlinear optical waveguides withvariable Kerr coefficient along the propagation distance.FKA acknowledges the European Community for twoyears grant PIIF-GA-2009-236099. PGK acknowledgessupport from NSF-DMS-0349023, NSF-DMS-0806762and the A. von Humboldt Foundation. MS thanks theMIUR for support through a PRIN-2008 initiative. [1] S. Flach and A.V. Gorbach, Phys. Rep. , 564(1993); P. Rosenau, Phys. Rev. Lett Soliton management in periodic systems ,Springer-Verlag (Berlin, 2007).[4] N.J. Smith et al. , Electron. Lett. , 54 (1996); I.Gabitov, S.K. Turytsin, Optics Lett. , 327 (1996).[5] H. Saito and M. Ueda, Phys. Rev. Lett.
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