Motion of discrete solitons assisted by nonlinearity management

Motion of discrete solitons assisted by nonlinearity management Jesús Cuevas,1Boris A. Malomed,2and P. G. Kevrekidis3 1Grupo de Física No Lineal, Departamento de Física Aplicada I, Escuela Universitaria Politécnica, C/ Virgen de África, 7, 41011 Sevilla, Spain 2Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 3Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA 共Received 12 November 2004; published 29 June 2005兲 We demonstrate that time-periodic modulation of the nonlinearity coefficient in the discrete nonlinear Schrödinger equation strongly facilitates creation of traveling solitons in the lattice. We predict this possibility in a semiqualitative form analytically, and test it in direct numerical simulations. Systematic computations reveal several generic dynamical regimes, depending on the amplitude and frequency of the time modulation, and on the initial thrust which sets the soliton in motion. These regimes include irregular motion of the soliton, regular motion of a decaying one, and regular motion of a stable soliton. The motion may occur in both the straight and reverse directions, relative to the initial thrust. In the case of stable motion, extremely long simulations in a lattice with periodic boundary conditions demonstrate that the soliton keeps moving indefinitely long without any visible loss. Velocities of moving stable solitons are in good agreement with the analytical prediction, which is based on requiring a resonance between the ac drive and motion of the soliton through the periodic lattice. The generic dynamical regimes are mapped in the model’s parameter space. Collisions between moving stable solitons are briefly investigated too, with a conclusion that two different outcomes are possible: elastic bounce, or bounce with mass transfer from one soliton to the other. The model can be realized experimentally in a Bose-Einstein condensate trapped in a deep optical lattice. DOI: 10.1103/PhysRevE.71.066614 PACS number共s兲: 05.45.Yv, 42.65.Wi, 05.45.⫺a, 03.75.Lm I. INTRODUCTION The discrete nonlinear Schrödinger 共DNLS兲equation is a well-known model of nonlinear lattice dynamics, which allows to study many generic features of nonintegrable dynamics 关1兴. This equation also finds direct applications to arrays of nonlinear optical waveguides 共as it was predicted long ago 关2兴and demonstrated in detail more recently, see Refs. 关3兴 and references therein兲, and to arrays of droplets in BoseEinstein condensates 共BECs兲trapped in a very deep optical lattice 关4兴. In all these contexts, discrete solitons are fundamental dynamical excitations supported by the DNLS equation. The dynamics of standing solitons, pinned by the underlying lattice, is understood quite well, in terms of both numerical simulations and analytical approximations, the most general one being based on the variational method 关5兴. However, moving discrete solitons pose a much more complex issue 关6,7兴. While, strictly speaking, exact solutions for moving solitons should not exist because of the radiation loss, direct simulations indicate that a soliton may move freely if its “mass” 共quadratic norm兲does not exceed a certain critical value 关7兴. In the quasicontinuum approximation, the source of the braking force acting on the moving soliton is the effective Peierls-Nabarro 共PN兲potential induced by the lattice 关8兴. In the case of the DNLS equation describing an array of nearly isolated droplets of a BEC in a deep optical lattice, an interesting possibility is to apply the ac Feshbach-resonance management 共FRM兲to it, as it was recently proposed in Ref. 关9兴. FRM may be induced by an external ac magnetic field, which periodically 共in time兲changes the sign of the nonlinearity by dint of the Feshbach resonance affecting collisions between atoms 共for a one-dimensional BEC without the optical lattice, the concept of FRM was elaborated in Ref. 关10兴兲. In this work, our aim is to demonstrate that the FRM, applied to the DNLS model, can strongly facilitate the motion of discrete solitons, which is a notable dynamical effect in the discrete systems 共Ref. 关9兴was only dealing with standing ones兲. The corresponding DNLS equation is iu ˙n+un+1 +un−1 −2un+g共t兲兩un兩2un=0, 共1兲 where un共t兲are the BEC wave functions at the lattice sites, and the real time-dependent nonlinear coefficient, proportional to the scattering length of the interatomic collisions 共with the sign minus兲,is g共t兲=gdc +gac sin共 ␻ t兲.共2兲 Note that gdc can be always chosen positive, because Eq. 共1兲 with gdc⬍0 can be transformed into that with gdc⬎0by means of the so-called staggering transformation un共t兲⬅共−1兲ne−4itu ˜ n共t兲.共3兲 The subsequent presentation is structured as follows: In Sec. II, using a Gaussian approximation for the soliton, we present an analytical estimate of the effective PN potential for the moving discrete soliton. The estimate suggests that the ac modulation of g共t兲may indeed help to suppress the PN potential, and thus facilitate free motion of discrete solitons. In Sec. III, we display results of systematic simulations, summarized in the form of diagrams in the parameter plane PHYSICAL REVIEW E 71, 066614 共2005兲 1539-3755/2005/71共6兲/066614共8兲/$23.00 ©2005 The American Physical Society066614-1 共 ␻ ,gac兲. The diagrams feature several generic dynamical regimes, including a large area of stable progressive motion. Collisions between solitons moving in opposite directions are briefly considered too, with a conclusion that they bounce from each other, sometimes featuring mass transfer between the solitons. Diverse dynamical regimes predicted in this work suggest straightforward possibilities for new experiments in BECs trapped in deep optical lattices, as discussed in the concluding Sec. IV which summarizes the findings reported in this paper. II. ANALYTICAL APPROXIMATION The continuum limit suggests the following ansatz for a moving soliton 关11兴, uans共n,t兲=Aexp 冉 −a关n− ␰ 共t兲兴2+i ␾ 共t兲+共i/2兲 ␰ ˙n −共i/4兲 冕 关 ␰ ˙共t兲兴2dt 冊 ,共4兲 where A,a, ␰ 共t兲, and ␾ are, respectively, the amplitude, squared inverse width, center position, and phase of the soliton. Accordingly, ␰ ˙is the soliton’s velocity, ␰ ˙/2 simultaneously being the wave number of the wave field carrying the moving soliton. For the soliton in the continuum NLS equation, the variational approximation yields the following relations: ␾ ˙=3a,A2=4 冑2a/g,共5兲 if g=const⬎0. In Eqs. 共5兲,ais regarded as an arbitrary positive constant 共intrinsic parameter of the soliton family兲. The only approximation which can produce a tractable result assumes a quasicontinuum ansatz for the soliton in the discrete system. Following this approach, we use the Hamiltonian corresponding to the DNLS equation 共1兲: H=兺 n=−⬁ +⬁ 冉 2兩un兩2−共un *un+1 +unun+1 *兲−g 2兩un兩4 冊 ,共6兲 the asterisk standing for the complex conjugation, and substituting into it the ansatz 共4兲. This way, the potential of the soliton-lattice interaction is obtained in the form of a Fourier series, H共 ␰ , ␰ ˙兲=兺m=0 ⬁Hm共 ␰ ˙兲cos共2 ␲ m ␰ 兲. In the case of a broad soliton, for which the ansatz 共4兲is relevant 共or, in a more general case, just to have the approximation in a tractable form兲, we keep only the lowest harmonic 共m=1兲in this expression, which is nothing else but the PN potential UPN. After some straightforward algebra 共using the Poisson summation formula兲, we thus find UPN共 ␰ , ␰ ˙兲=1 2冑 ␲ aA2exp 冉 − ␲ 2 4a 冊 冋 4冑2 exp 冉 − ␲ 2 4a 冊 ⫻关1+e−a/2 cos共 ␰ ˙/2兲兴 −gA2冑 ␲ a 册 cos共2 ␲␰ 兲. 共7兲 A noteworthy feature in this estimate, in comparison with known perturbative results for static solitons 关8兴,is the dependence on the potential’s amplitude on the soliton’s velocity, ␰ ˙. If the relation between aand A2for the soliton in the continuum NLS equation, as given by Eq. 共5兲共for constant g兲, is substituted in Eq. 共7兲, the coefficient in front of cos共2 ␲␰ 兲, i.e., the amplitude of the PN potential, never vanishes. However, it may vanish if the underlying pulse 共4兲is considered not as a soliton, but just as a pulse with independent amplitude and inverse-width parameters A2and a; then, the condition of the vanishing of the PN potential determines adiscrete spectrum of the velocities ␰ ˙, in the form 1 + exp共−a/2兲cos共 ␰ ˙/2兲=共gA2/4兲冑 ␲ /共2a兲exp关 ␲ 2/共4a兲兴, 共8兲 provided that gA2is small enough to make the right-hand side of Eq. 共8兲smaller than 2 共this caveat is essential, as the factor exp关 ␲ 2/共4a兲兴 may be exponentially large兲. In this work, however, our objective is not to verify this possibility, but rather to consider the case when the nonlinear coefficient gis a function of time, as defined in Eq. 共2兲. Note FIG. 1. Position of the center of mass Xas a function of time for a typical example of a dynamical regime in which 共a兲the soliton remains pinned 共gac=0.03, ␻ =1, and q=0.5兲and 共b兲the soliton develops an irregular motion 共gac=0.065, ␻ =1, and q=0.5兲. CUEVAS, MALOMED, AND KEVREKIDIS PHYSICAL REVIEW E 71, 066614 共2005兲 066614-2 that, for a broad soliton 共small a兲, the PN potential barrier is exponentially small, hence the soliton’s kinetic energy may be much larger than the potential. This implies that the velocity of the soliton moving through the potential 共7兲with the period L=1 contains a constant 共dc兲part and a small ac correction to it, with the frequency 2 ␲␰ ˙0/L⬅2 ␲␰ ˙0关12兴, ␰ ˙共t兲⬇ ␰ ˙0+ ␰ ˙1cos共2 ␲␰ ˙0t兲, ␰ ˙1 2Ⰶ ␰ ˙0 2.共9兲 Then, substituting the expression 共9兲into the condition 共8兲, which provides for the suppression of the PN potential, one can expand its left-hand side, 1 + exp共−a/2兲cos共 ␰ ˙/2兲⬇1 + exp共−a/2兲关cos共 ␰ ˙0/2兲 −共 ␰ ˙1/2兲sin共 ␰ ˙0/2兲cos共2 ␲␰ ˙0t兲兴. 共10兲 Now, inserting the variable g共t兲from Eq. 共2兲into the righthand side of Eq. 共8兲, it is obvious that gdc and gac can be chosen so as to secure this equation to hold, provided that the average soliton’s velocity takes the resonant value, ␰ ˙0 = ␻ /2 ␲ . More generally, due to anharmonic effects, one may expect the existence of a spectrum of the resonant velocities, ␰ ˙0=共cres兲N 共M兲⬅M ␻ 2 ␲ N,共11兲 with integers Mand N. Actually, an ac drive can support stable progressive motion of solitons at the resonant velocities 共11兲共assuming the spatial period L=1兲, even in the presence of dissipation, in a broad class of systems. This effect was first predicted for discrete systems 共of the Toda-lattice and Frenkel-Kontorova types兲in Ref. 关13兴, and demonstrated experimentally in an LC electric transmission line in Ref. 关14兴. Later, the same effect was predicted 关15兴and demonstrated experimentally 关16兴in continuous long Josephson junctions with a spatially periodic inhomogeneity. However, a qualitative difference of the present prediction is that it applies to nontopological solitons, while all the previously known examples involved kinks, i.e., discrete of continuum solitons whose topological charge directly couples to the driving field 关this is why the above approximation, even if it is a crude one, is relevant, as the relation 共11兲, established for the ac-driven kinks, cannot be immediately applied to nontopological solitons兴. The only example similar to what is suggested here in the context of nontopological solitons that we are aware of, was reported in Ref. 关17兴共the model involved spatially uniform dissipation and an ac drive localized in space兲. Note also that the mechanism of the ac-driven motion considered here is different from that in ratchet systems 共see, e.g., Ref. 关18兴and referFIG. 2. An example of asymmetric splitting of the soliton, for gac=0.196, ␻ =0.5, and q=0.5. Panel 共a兲shows the position of two local density maxima corresponding to the secondary solitons 共splinters兲, and panel 共b兲shows the global evolution of the lattice field. FIG. 3. A generic example of the progressive motion of a decaying soliton in the straight direction, for gac=0.206, ␻ =0.5, and q=0.5. MOTION OF DISCRETE SOLITONS ASSISTED BY …PHYSICAL REVIEW E 71, 066614 共2005兲 066614-3 ences therein兲, as the DNLS equation does not break the mirror symmetry, n−n. III. NUMERICAL RESULTS We now proceed to direct simulations of the system discussing some of the relevant theoretical predictions in light of the numerical results. We integrate Eq. 共1兲with an initial configuration in the form of a standing-soliton solution for gac=0. This solution has the form of un共t兲=vnexp共i ␹ t兲, with the real field vnobeying the equation ␹ vn=vn+1 +vn−1 −2 vn+gdcvn 3.共12兲 Equation 共12兲was solved by means of well-known methods 共starting from the anticontinuum limit兲. Then, to set the soliton in motion, it was given a lattice momentum q, so that the initial condition was un共0兲=vnexp共inq/2兲.共13兲 Equation 共13兲implies that the soliton will move to the right if q⬎0. The results will be displayed for gdc=1, ␹ =1, and three different values of the initial thrust, q=0.25, q=0.5, and q =1, as these cases were found to represent a generic situation in the plane of the ac-drive’s parameters 共 ␻ ,gac兲. Simulations were run in the time interval 0⬍t⬍100⫻共2 ␲ / ␻ 兲or longer, by means of the fourth-order Runge-Kutta algorithm, with the time step ⌬t=0.002. In most cases, the lattice with 251 sites was used. Edge absorbers were installed by adding the loss term i ␥ un, with ␥ =1,toEq.共1兲at the 10 sites adjacent to each edge. Besides that, in some cases extremely long simulations were performed in a larger lattice with periodic boundary conditions, to verify if the stable motion of the soliton could last for very long times, and also to examine collisions between the solitons, see below. If gac=0, the soliton pushed as per Eq. 共13兲with qⱗ0.7 does not start progressive motion. Instead, it remains pinned to the lattice, with its center oscillating around an equilibrium position. This observation may be explained by the fact that the kinetic energy given to the soliton is smaller than the height of the PN potential barrier. Several distinct types of dynamics were observed with gac⬎0, depending on the driving frequency ␻ and the thrust parameter q. First, the soliton may remain pinned, as shown in Fig. 1共a兲. In this case, the simulations 关run in the interval 0⬍t⬍300共2 ␲ / ␻ 兲兴 demonstrate that the soliton stays pinned within a few sites from its initial position. The central coordinate, the evolution of which is displayed in Fig. 1 and other figures, is defined as X= 兺 n n兩un兩2 兺 n 兩un兩2.共14兲 The next generic regime is that of irregular motion, as shown in Fig. 1共b兲. A characteristic feature of this regime is FIG. 4. The same as in Fig. 4, in the case of motion in the reverse direction, for gac=0.170, ␻ =1, and q=0.5. FIG. 5. A generic example of the motion of a stable 共nondecaying兲soliton in the straight direction, for gac=0.132, ␻ =1, and q=0.5. CUEVAS, MALOMED, AND KEVREKIDIS PHYSICAL REVIEW E 71, 066614 共2005兲 066614-4 that the soliton randomly changes the direction of motion several times, and the velocity remains very small in comparison with regimes of “true motion,” see below. Under the action of a strong drive, the soliton can sometimes split into two secondary ones moving in opposite directions, see Fig. 2. As can be observed, the splitting is strongly asymmetric, and the heavier secondary soliton 共splinter兲may move both forward and backward, relative to the initial push. We notice that splitting of a quiescent soliton 共without any initial thrust applied to it兲into two symmetric splinters, moving in opposite directions with equal velocities, was reported, in the same model based on Eq. 共1兲, in Ref. 关9兴. The initial push applied to the soliton is a natural cause for the symmetry breaking observed here in the case of the splitting. In the case of a moderately strong drive, the splitting does not occur 共see Fig. 9 below兲. Instead, two distinct generic regimes of regular motion of the soliton were observed. In one case, shown in Figs. 3 and 4, the moving soliton is not really stable, as it gradually decays into radiation. The most interesting case is that of persistent motion of a stable soliton, without any observable decay 共after an initial transient stage of the evolution兲. Examples of the latter are displayed in Figs. 5 and 6. To distinguish between the two different regimes of motion, we have adopted a criterion that the moving soliton is stable if it keeps more than 70% of its norm, 兺兩un兩2, in the core. It was found that the criterion in this form makes it possible to accurately identify truly stable regimes, if the simulation is extended for a much longer time 共see below兲. Typical examples displayed in Figs. 3–6 demonstrate that free motion of the soliton is possible in both the straight and reverse directions, relative to the initial thrust. As stable motion of solitons in nonintegrable discrete models is an issue of theoretical and experimental interest, we have further investigated this case, replacing the finite lattice with edge absorbers by a ring-shaped one, with periodic boundary conditions. This setting opens a way to study indefinitely long motion of the soliton. The result, illustrated by examples shown in Fig. 7, is that the moving solitons which were identified as stable by the above criterion, remain stable indeed 共preserving their shape兲as long as the simulations could be run. In this case, it is pertinent to compare the average velocity c ¯ of the persistent motion with the prediction given by Eq. 共11兲. The result is c ¯ 1⬇0.246 and c ¯ 2⬇0.155 in the cases shown in Figs. 7共a兲and 7共b兲, respectively. Comparison with the analytical formula 共11兲共with ␻ =1, which is the driving frequency in the examples shown in Fig. 7兲demonstrates that c ¯ 2and c ¯ 1fit well to the predicted values in the cases of the, respectively, fundamental and second-order resonance, c ¯ 2/共cres兲1 共1兲⬇0.974, c ¯ 1/共cres兲3 共2兲⬇1.029. 共15兲 Relatively small discrepancies between the predicted and observed values in Eq. 共15兲can be accounted for by the fact FIG. 6. The same as in Fig. 6, but in the case of the motion of a stable soliton in the reverse direction, for gac=0.122, ␻ =1, and q=0.5. FIG. 7. Indefinitely long motion of stable solitons in a ring lattice composed of 601 sites. The gray-scale plots show the spatiotemporal distribution of the density, 兩un共t兲兩2. The examples of the straight 共a兲and inverse 共b兲motion pertain, respectively, to gac=0.132, ␻ =1, q=0.5, and gac=0.122, ␻ =1, q=0.5. MOTION OF DISCRETE SOLITONS ASSISTED BY …PHYSICAL REVIEW E 71, 066614 共2005兲 066614-5 that the effective perturbations are not really weak in these cases. The soliton adjusting itself to the stable-motion mode typically sheds off about 20% of its initial norm. This conspicuous amount of radiation does not essentially affect the established regime. In the cases shown in Figs. 3–6 the emitted radiation is partly suppressed by the edge absorbers. However, even in the case displayed in Fig. 7, when all the radiation stays in the lattice subject to the periodic boundary conditions, its presence does not give rise to any appreciable perturbation in the long-time motion of the soliton. In fact, this observation is an additional essential evidence of the robustness of the moving soliton in the corresponding regime. FIG. 8. 共Color online兲Maps in the left column show areas in the parameter plane 共 ␻ ,gac兲which give rise to the following dynamical regimes. The rows from top to bottom correspond to values 0.25, 0.5, and 1 of the initial-push parameter q. White areas: the soliton remains pinned; cyan: irregular motion; green: splitting; magenta: regular motion with decay; black: stable motion 共without decay兲. The maps in the right column additionally show the difference between the forward 共alias straight, marked by magenta兲and backward 共alias reverse, marked by black兲directions of the regular motion, relative to the direction singled out by the initial push. Regular-motion regimes for both decaying and stable solitons are included here. CUEVAS, MALOMED, AND KEVREKIDIS PHYSICAL REVIEW E 71, 066614 共2005兲 066614-6 For the fixed values of gdc=1 in Eq. 共2兲and ␹ =1 in Eq. 共13兲, and several values of the initial thrust in Eq. 共13兲, q=0.25, 0.5, and 1, we have collected results of systematic simulations, varying the drive’s parameters, gac and ␻ ,by small steps in broad ranges spanning the two-parameter space. The results are summarized in Fig. 8, in the form of maps in the 共 ␻ ,gac兲plane, where we outline regions giving rise to each of the qualitatively different dynamical regimes described above. This map is the key finding of the present work, detailing the various possibilities arising as a result of the application of the FRM to the lattice soliton. Some general features can be deduced from the examination of the maps in Fig. 8. The increase of the initial thrust q significantly affects the map, although chiefly quantitatively, rather than qualitatively. At all values of q, the irregular motion is, generally, changed by stable progressive motion 共straight or reverse兲with the increase of the drive’s amplitude, and/or decrease of its frequency, which seems quite natural. Further increase of the drive’s strength, which implies the applications of a strong perturbation to the system, may be expected to lead to an instability, which indeed happens, in the form of onset of the gradual decay of the moving solitons. Finally, strong instability sets in, manifesting itself in the splitting of the soliton. It also seems natural that the soliton is more prone to splitting if the driving frequency is low, as internal strain in the pulse, which eventually leads to its splitting, has more time to accumulate if the drive oscillates slowly. Transition to the reverse motion tends to happen parallel to the transition from the stable moving soliton to the decaying one. For this reason, in most cases 共but not always兲 backward-moving solitons are decaying ones. Finally, it should be noted that the increase of the initial thrust leads to overall stabilization of the soliton 共somewhat counterintuitively兲, making the decay and splitting zones smaller. Finally, using the large lattice with the periodic boundary conditions, we also simulated collisions between solitons originally moving with opposite velocities. Initial pulses were generated by applying the thrust ±qto two quiescent solitons. A systematic study of collisions is very difficult in the present model, cf. Ref. 关7兴. Nevertheless, we were able to identify two different types of the interaction, typical examples of which are shown in Fig. 9. In the case of Fig. 9共a兲, the solitons bounce back from each other elastically. Afterwards, one of the solitons spontaneously reverses its direction of motion, due to its interaction with the underlying lattice. Eventually, we observe a pair of weakly interacting solitons traveling for long times in the same direction. In another case, Fig. 9共b兲, the solitons also bounce after the first collision; however, in this case the collision is inelastic, resulting in transfer of mass 共norm兲from one soliton to the other. Repeated collisions lead to additional transfer, and eventually the weak soliton almost disappears. It is relevant to note that, in contrast to what is known about collisions between moving solitons in the ordinary DNLS equation 共the one with constant coefficients兲关7兴, we have never observed merger of colliding solitons into a standing one. IV. CONCLUSIONS In this work, we have investigated moving nontopological solitons in the DNLS equation with a periodically timemodulated nonlinear coefficient. An approximate analytical consideration predicts that the ac nonlinearity management may support stable traveling solitons in the lattice. Systematic simulations reveal several generic dynamical regimes, depending on parameters of the time modulation, and the size of the initial thrust which sets the soliton in motion. Besides the possibility that the soliton remains pinned, or is split by a very strong drive, the basic dynamical regimes feature irregular motion, regular motion of a decaying soliton, and regular motion of a stable one. In the latter case, extremely long simulations in the lattice with periodic boundary conditions demonstrate that the soliton keeps moving indefinitely long without any tangible loss. Velocities of the moving stable solitons are found to be in good agreement with the analytical prediction through a resonance condition, a noteworthy fact being that the spectrum of the resonant velocities for the nontopological solitons is the same as for topological ones 共kinks兲, although the dynamical mechanisms supporting the progressive motion in the two cases are quite different. All the generic dynamical regimes were mapped in the model’s parameter space. Collisions between stable moving solitons were briefly investigated too, with a conclusion that two different outcomes are possible, elastic bounce, and bounce with mass transfer between the solitons. The model can be implemented in a quasi-onedimensional 共“cigar-shaped”兲Bose-Einstein condensate segFIG. 9. Two typical examples of different outcomes of collisions between solitons with equal masses moving in opposite directions in the lattice with periodic boundary conditions. The parameters are gac=0.132, ␻ =1, q=0.5 共a兲and gac=0.122, ␻ =1, q=0.5 共b兲. Note that the collision is multiple in panel 共b兲. MOTION OF DISCRETE SOLITONS ASSISTED BY …PHYSICAL REVIEW E 71, 066614 共2005兲 066614-7 mented by a strong optical lattice, the nonlinearity modulation induced by the Feshbach resonance in the ac regime. In this case, a realization is possible with ⬃103atoms trapped in each of a few filled local potential wells. The setting may be essentially the same as in the recent experimental work which has resulted in the creation of one-dimensional gap solitons 关19兴and observation of the nonlinear self-trapping 关20兴in the repulsive condensate 共without the Feshbach management兲. The fact that the repulsive condensate 共87Rb兲was used in the experiments is not an obstacle, in view of the applicability of the staggering transformation of Eq. 共3兲. 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