Discrete Nonlinear Schrödinger Equations with Time-Dependent Coefficients (Management of Lattice Solitons)

Chapter15 Discrete Nonlinear Schr¨ odinger Equations with Time-Dependent Coefficients (Management of Lattice Solitons) Jes´us Cuevas and Boris A. Malomed 15.1 Introduction The general topic of the book into which this chapter is incorporated is the discrete nonlinear Schr¨odinger (DNLS) equation as a fundamental model of nonlinear lattice dynamics. The DNLS equation helps to study many generic features of nonintegrable dynamics in discrete media [1]. Besides being a profoundly important model in its own right, this equation has very important direct physical realizations, in terms of arrays of nonlinear optical waveguides (as was predicted long ago [2] and demonstratedin detail more recently, see [3, 4] and referencestherein), and arrays of droplets in Bose–Einstein condensates (BECs) trapped in a very deep optical lattice (OL), see details in the original works [5–10] and the review [11]. In all these contexts, discrete solitons are fundamental localized excitations supported by the DNLS equation. As explained in great detail in the rest of the book, the dynamics of standing solitons, which are pinned by the underlying lattice, is understood quite well, by means of numerical methods and analytical approximations (the most general approximation is based on the variational method [12, 13]). A more complex issue is posed by moving discrete solitons [14–17]. While, strictly speaking, exact solutions for moving solitons cannot exist in nonintegrable lattice models because of the radiation loss, which accompanies their motion across the lattice, direct simulations indicate that a soliton may move freely if its norm (“mass”) does not exceed a certain critical value [17]. In the quasi-continuum approximation, the moving soliton may be considered, in the lowest (adiabatic) approximation, as a classical mechanical particle which moves across the effective Peierls–Nabarro (PN) potential induced by the lattice [18–21]. In this limit, the radiation loss is a very weak nonadiabatic effect, which attests to the deviation of the true soliton dynamics from that of the point-like particle. In the case of the DNLS equation describing arrays of nearly isolated droplets of a BEC trapped in a deep OL, an interesting possibility is to apply the Feshbach J. Cuevas (B) Grupo de F´ısica No Lineal, Departamento de F´ısica Aplicada I, Escuela Universitaria Polit´ecnica, C/ Virgen de ´ Africa, 7, 41011 Sevilla, Spain e-mail: [email protected] P.G. Kevrekidis, The Discrete Nonlinear Schr¨odinger Equation, STMP 232, 277 277–291, DOI 10.1007/978-3-540-89199-4 15, c Springer-Verlag Berlin Heidelberg 2009 278 J. Cuevas and B.A. Malomed resonance management (FRM) to this system, as was first proposed and studied for immobile discrete solitons in [22], and later elaborated in detail, for moving solitons, in [23]. The FRM may be induced by an external low-frequency ac magnetic field, which periodically (in time) changes the sign of the nonlinearity by dint of the FRM, i.e., formation of quasi-bound states in collisions between atoms [24]. For the effectively one-dimensional BEC in the absence of the OL, the concept of the FRM was elaborated in (1D) [25], see also the book [26]. The objective of the chapter is to summarize basic results for the quiescent and moving 1D DNLS solitons subjected to the time-periodic management, following, chiefly, the lines of works [22] and [23]. For the quiescent solitons, most significant findings are FRM-induced resonances in them, and stability limits for the solitons which are affected by the resonant mechanisms. In particular, resonances with an external time-periodic modulation may stimulate self-splitting of the solitons. For the moving solitons, an essential conclusion is that the FRM may strongly facilitate their mobility, which is an essentially novel dynamical effect in discrete media. The DNLS equation which includes the FRM mechanism can be cast in the following form: i˙un+C(un+1+un−1−2un)+g(t)|un|2un=0,(15.1) where un(t) denotes the BEC wave function at the lattice sites, Cis the strength of the linear coupling between adjacent sites of the lattice, and the real time-dependent nonlinear coefficient is g(t)=gdc +gac sin(ωt) (15.2) with the time-dependent term accounting for the FRM (−gis proportional to the scattering length of atomic collisions, whose magnitude and sign may be directly altered by the FRM). In what follows below, we fix, by means of obvious rescaling, C≡1. We also note that gdc may always be chosen positive, as it can be transformed by means of the so-called staggering transformation, un(t)≡(−1)ne−4it ˜un(t). Equation (15.1) with the time-dependent nonlinear coefficient has a single dynamical invariant, the norm (which is proportional to the number of atoms in the BEC), N=+∞  n=−∞|un|2.(15.3) The chapter is divided into two major parts that deal with quiescent and mobile discrete solitons (the latter one also briefly considers collisions between the moving solitons). Each part contains sections which present analytical and numerical results. In either case, the analytical approach is based on using a particular ansatz for the shape of the soliton, while numerical results are produced by systematic direct simulations of Eq. (15.1) with appropriate initial conditions. 15 DNLS with Time-Dependent Coefficients 279 15.2 Quiescent Solitons Under the Action of the “Management” 15.2.1 Semi-Analytical Approximation In this section, we put Eqs. (15.1) and (15.2) in a slightly different form, namely, i˙un+1 2(un+1+un−1−2un)+a(t)|un|2un=0,a(t)=1+a1sin (ωt),(15.4) where, as said above, we fix C=1, and the dc part of a(t)isalsosetequalto1 by means of an additional rescaling. A (semi-)analytical approximation for soliton solutions to Eq. (15.4) is based on the fact that it can be derived from the Lagrangian, L=1 2 ∞  n=−∞iu∗ n˙un−un˙u∗ n−|un+1−un|2+a(t)|un|4(15.5) (∗stands for the complex conjugate). Then, the variational approximation (VA) [13] represents the solitons by the following ansatz, following [12]: un(t)=Aexp(iφ+ib|n|−α|n|),(15.6) where A,φ,b,andαare real functions of time. Substituting this ansatz in the Lagrangian (15.5), it is easy to perform the summations explicitly and thus arrive at the corresponding effective Lagrangian (an inessential term, proportional to ˙ φ,is dropped here): L N=− 1 sinh (2α) db dt +cosb coshα+1 4Na(t)sinh α cosh3αcosh(2α),(15.7) where N=A2coth αis the norm of the ansatz (recall the norm is the dynamical invariant, calculated as per Eq. (15.3)). The variational equations for the soliton’s chirp b and inverse width α, derived from Lagrangian (15.7) are db dt =2(cosb)sinh3α cosh(2α)−1 2Na(t)tanh2α2cosh(2α)−1 cosh(2α),(15.8) dα dt =−(sin b)(sinhα)tanh(2α) (15.9) (the amplitude Awas eliminated here in favor of α, due to the conservation of N). First, in the absence of the FRM, i.e., for a(t)=const ≡1 (see Eq. (15.4)) Eqs. (15.8) and (15.9) give rise to a stationary solution (fixed point, FP), with bFP =0andαFP defined by equation sinh(αFP)=N1+3tanh 2(αFP)/4[12]. The VA-predicted stationary soliton is quite close to its counterpart found from a numerical solution of Eq. (15.4) with a(t)≡1 [22]. Furthermore, linearization of 280 J. Cuevas and B.A. Malomed Eqs. (15.8) and (15.9) around the FP yields a squared frequency of intrinsic oscillations for a slightly perturbed soliton, ω2 0=sinh3(αFP)cosh 2(αFP) cosh3(2αFP)4sinh(αFP)[cosh(2αFP)+2] −N cosh4(αFP)5cosh 2(2αFP)−2cosh(2αFP)−1(15.10) (this expression can be shown to be always positive). Comparison of this prediction of the VA with numerically found frequencies of small oscillations of the perturbed soliton demonstrates good agreement too [22]. In the presence of the FRM, a1= 0, strong (resonant) response of the system is expected when the modulation frequency ωis close to eigenfrequency given by Eq. (15.10). Moreover, the dynamics may become chaotic, via the resonanceoverlapping mechanism [27], if the modulation amplitude a1exceeds some threshold value. This was observed indeed in numerical simulations of Eqs. (15.8) and (15.9), as illustrated in Fig. 15.1 by a typical example of the Poincar´emap[27]in the chaotic regime. The figure shows discrete trajectories initiated by two sets of the initial conditions, namely, b(1) 0,α(1) 0=(0,0.789),that correspondto the stationary discrete soliton with amplitude A=1 (cf. ansatz (15.6)) in the unperturbed system (a1=0), and a different set, b(2) 0,α(2) 0=(0.13,0.74). The respective modulation frequency,ω, is close to the eigenfrequency of small oscillations ω0, as predicted by Eq. (15.10). For the former initial condition, the point in space (b,α) is chaotically movingaway from the unperturbedFP. However, the chaotic evolution is a transient feature, as the discrete trajectory takes an asymptotic form with α(t)→0, which implies decay (indefinite broadening) of the soliton. The second set of the initial conditions eventually leads to a stable periodic solution (in terms of the Poincar´e map, it is represented by a new FP, which is found in a vicinity of the unperturbed one). The latter results predicts the existence of quasi-stationary discrete solitons under the action of the FRM. Fig. 15.1 Example of the chaotic Poincar´emap generated by Eqs. (15.8) and (15.9) with a1=0.02766, ω=0.481, and W=1.5202. Reprinted from [22] with permission 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 –0.4 –0.2 0 0.2 0.4 α b 15 DNLS with Time-Dependent Coefficients 281 Another analytically tractable case is one for the high-frequency modulation. It is then possible to perform averaging of Eq. (15.4) (without resorting to the VA). The eventual result is an effective DNLS equation for the slowly varying part of the lattice field, qn(t)[22]: i˙qn+1 2(qn+1+qn−1−2qn)+|qn|2qn =a2 1/8ω23|qn|4(qn+1+qn−1)+2|qn|2q2 nq∗ n+1+q∗ n−1 +|qn+1|4qn+1+|qn−1|4qn−1−2|qn+1|22|qn|2qn+1+q2 nq∗ n+1 −2|qn−1|22|qn|2qn−1+q2 nq∗ n−1.(15.11) Equation (15.11) is the DNLS equation with a small inter-site quintic perturbation (a2 1/8ω2is a small perturbation parameter). 15.2.2 Direct Simulations Systematic direct simulations of Eq. (15.4) demonstrate that the VA correctly predicts only an initial stage of the dynamics [22]. Emission of linear waves (lattice phonons) by the soliton, which is ignored by the VA, gives rise to an effective dissipation, that makes the resonance frequency different from the value predicted by Eq. (15.10). Actually, the soliton decouples from the resonance, as ω0depends on the norm N, and the radiation loss results in a gradual decrease of the norm. Nevertheless, the general predictions of the VA turn out to be correct for a10.05: under the action of the management, oscillations of the soliton’s parameters are regular for very small a1, and become chaotic at larger a1. Typical examples of the soliton dynamics under the action of stronger management, with a1≥0.1(andω=0.5) are displayed in Fig. 15.2. A noteworthy observation, which could not be predicted by the single-soliton ansatz, is splitting of the pulse, which is observed, at a1=0.2, in Fig. 15.2, while at other values of a1, both smaller and larger than 0.2, the soliton remains centered around n=0. Note that the splitting is similar to that revealed by direct simulations of the continuum NLS equation with a term accounting for periodic modulation of the linear dispersion (dispersion management), which was reported in [28]. A similar phenomenon was also observed in a discrete model with the finite-difference dispersion term subject to periodic modulation [29]. Results of the systematic numericalstudy of the evolutionof solitons in Eq. (15.4) are summarized in Fig. 15.3 (for a10.2, the pulse may split into several moving splinters). The diagram shows that the actual critical value of management amplitude a1, past which the soliton develops the instability (via the splitting) is much higher than the prediction of the chaotic dynamics threshold by the VA (which also eventually leads to the decay of the soliton, as the chaotic transient is followed by the asymptotic stage of the evolution with α(t)→0, see above). Thus, the VA based on Eqs. (15.8) and (15.9) underestimates the effective stability of the discrete 282 J. Cuevas and B.A. Malomed 2 a1 = 0.1 a1 = 0.2 a1 = 0.3 –30–20–10 010 20 30 n0 50 100 t 0 0.5 1 1.5 u –30–20–10 010 20 30 n 0 50 100 t 0 0.5 1 1.5 –30–20–10 010 20 30 n 050 100 t 0 0.5 1 1.5 2 u 2 u Fig. 15.2 Evolution of a discrete soliton with initial amplitude A=1 in Eq. (15.4) with ω=0.5 and different values of the management amplitude, a1. Reprinted from [22] with permission solitons. This conclusion is explained by the fact that the radiation loss, that was ignored by the variational ansatz, plays a stabilizing role for the discrete solitons. It is worth mentioning that the existence of a finite critical value of the modulation amplitude, past which the splitting occurs, and the fact that the actual stability area for solitons is larger than predicted by the VA are qualitatively similar to features found in the above-mentioned dispersion-management model based on the continuum NLS equation [28]. 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ω a1 Fig. 15.3 The diagram of dynamical regimes in the plane of the FRM parameters, (ω,a1), as produced by systematic simulations of Eq. (15.4) with a0=1 and initial soliton’s amplitude A=1. Open and solid squares correspond to stable and splitting solitons, respectively. The solid line is the chaos-onset threshold predicted by the numerical solution of the variational equations (15.8) and (15.9). Reprinted from [22] with permission 15 DNLS with Time-Dependent Coefficients 283 15.3 Supporting Moving Solitons by Means of the “Management” 15.3.1 Analytical Approximation In this part, we switch to the notation adopted in Eqs. (15.1) and (15.2), but again with C≡1. The continuum limit (i.e., the ordinary NLS equation) suggests the following ansatz for a moving soliton [30]: u(n,t)=Aexp −b(n−ξ(t))2+iφ(t)+i 2˙ ξn−i 4˙ ξ(t)2dt,(15.12) where A,b,ξ(t),and φare, respectively, the amplitude, squared inverse width, central coordinate, and phase of the soliton. Accordingly, ˙ ξis the soliton’s velocity, ˙ ξ/2 simultaneously being the wave number of the wave field which carries the moving soliton. Note the difference of this ansatz, which emulates the VA for the solitons in the continuum NLS equation [30], from the above-mentioned ansatz (15.6), that was adopted for the essentially discrete model. In the framework of the continuum NLS equation, the VA for the ordinary solitons yields ˙ φ=3b,A2=4√2b g(15.13) (assuming that g=const >0), where bis treated as an arbitrary positive constant, i.e., intrinsic parameter of the soliton family. Then, the ansatz (15.12) with zero velocity, ˙ ξ=0, may be substituted in the Hamiltonian corresponding to the DNLS equation (15.1), H=+∞  n=−∞ (2|un|2−u∗ nun+1+unu∗ n+1−g 2|un|4).(15.14) In this way, an effective potential of the soliton-lattice interaction is obtained in the form of a Fourier series, H(ξ)=∞ m=0Hmcos(2πmξ). In the case of a broad soliton, for which ansatz (15.12) is relevant, it is sufficient to keep only the lowest harmonic (m=1) in this expression, which yields the respective PN (Peierls–Nabarro) potential, UPN. A straightforward calculation, using the Poisson summation formula, yields [23] UPN(ξ)=1 2π bA2exp −π2 4b4√2exp−π2 4b 1+e−b/2−gA2π bcos(2πξ).(15.15) 284 J. Cuevas and B.A. Malomed If the relation between band A2, taken as for solitons in the continuum NLS equation, i.e., as per Eq. (15.13), is substituted into Eq. (15.15), the coefficient in front of cos(2πξ), i.e., the amplitude of the PN potential, never vanishes. However, it may vanish if (15.12) is considered not as a soliton, but just as a pulse with independent amplitude and width, Aand 1/√b; then, the PN potential in Eq. (15.15) may vanish, under the following condition: 1+exp −b 2=gA2 4π 2bexp π2 4b.(15.16) The vanishing of the PN potential implies a possibility of unhindered motion of the soliton across the lattice. For a broad soliton (small b), the PN potential barrier is exponentially small, hence, the soliton’s kinetic energy may be much larger than the height of the potential barrier. Therefore, the velocity of the soliton moving through the potential (15.15) with period L=1 contains a constant (dc) part and a small ac correction to it, with frequency 2π˙ ξ0/L≡2π˙ ξ0[31]: ˙ ξ(t)≈˙ ξ0+˙ ξ1cos2π˙ ξ0t,˙ ξ2 1˙ ξ2 0. Substituting this into condition (15.16), one can expand its left-hand side by using exp −b 2cos˙ ξ 2≈exp −b 2cos ˙ ξ0 2− ˙ ξ1 2sin ˙ ξ0 2cos2π˙ ξ0t. (15.17) Next, inserting the variable nonlinearity coefficient (15.2) into the right-hand side of Eq. (15.16), and equating the resulting expression to that (15.17), one concludes that gdc and gac may be chosen so as to secure condition (15.16) to hold, provided that the average soliton’s velocity takes the resonant value,˙ ξ0=ω/2π.Moregenerally, due to anharmonic effects, one may expect the existence of a spectrum of the resonant velocities, ˙ ξ0=(cres)(M) N≡Mω 2πN(15.18) with integers Mand N. Actually, an ac drive can support stable progressive motion of solitons at resonant velocities (15.18) (assuming that the spatial period is L=1),eveninthe presence of dissipation, in a broad class of systems. This effect was first predicted for discrete systems (of the Toda lattice and Frenkel–Kontorova types) in [32–35], and demonstrated experimentally in an LC electric transmission line [36]. Later, the same effect was predicted [37] and demonstrated experimentally [38] in long Josephson junctions with a spatially periodic inhomogeneity. However, a qualitative difference of the situation considered here is that we are now dealing with nontopological solitons, while the above-mentioned examples involved kinks, i.e., discrete or continuum solitons whose topologicalcharge directly couples to the driving field. 15 DNLS with Time-Dependent Coefficients 285 15.3.2 Numerical Results Numerical simulations of Eq. (15.1) (with C=1) were performed in [23], setting gdc =1 by means of the same rescaling which was used in the studies of the quiescent solitons. First, stationary solitons were found as solutions to the DNLS with gac =0,in the form of u(0) n(t)=vnexp(−iω0t). Then, the FRM with gac >0 was switched on, and, simultaneously, the soliton was set in motion by giving it a kick, i.e., multiplying unby exp(iqn). Generic results in the plane of the ac-drive’s parameters, (ω, gac), can be adequately represented by fixing ω0=−1 and considering three values of the kick, q=0.25, 0.5, and 1. If gac =0, the kicked soliton does not start progressive motion if the thrust is relatively weak, q0.7. It remains pinned to the lattice, oscillating around an equilibrium position, which may be explained by the fact that the kinetic energy imparted to the soliton is smaller than the height of the PN potential barrier. Several types of dynamics can be observed with gac >0, depending on the modulation frequency ωandkickstrengthq. First, the soliton may remain pinned (generally, not at the initial position, but within a few sites from it, i.e., the soliton passes a short distance and comes to a halt, as shown in Fig. 15.4a). The next generic regime is that of irregular motion, as illustrated in Fig. 15.4b. A characteristic feature of that regime is that the soliton randomly changes the direction of motion several times, and the velocity remains very small in comparison with regimes of persistent motion, see below. The soliton’s central coordinate, the evolution of which is presented in Fig. 15.4, is defined as X=nn|un|2/N, with the norm Ncalculated as per Eq. (15.3). Under the action of strong modulation, the soliton can sometimes split into two pulses moving in opposite directions, see Fig. 15.5. This outcome is similar to that found for quiescent solitons, cf. Fig. 15.2b, although the splitting of the kicked soliton is strongly asymmetric (unlike the nearly symmetric splitting of the quiescent 0 200 400 600 −4 0 4 8 12 (b) t X 0 500 1000 1500 −2 −1 0(a) t X Fig. 15.4 The soliton’s central position as a function of time, for typical cases in which the soliton remains pinned (a),atgac =0.03, or develops an irregular motion (b),atgac =0.065. In both cases, ω=1andq=0.5. Reprinted from [23] with permission