On the Modulational Instability of the Nonlinear Schrödinger Equation with Dissipation Z Rapti

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2004 On the Modulational Instability of the Nonlinear Schrödinger Equation with Dissipation Z Rapti

PG Kevrekidis University of Massachusetts - Amherst, [email protected]

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Recommended Citation Rapti, Z and Kevrekidis, PG, "On the Modulational Instability of the Nonlinear Schrödinger Equation with Dissipation" (2004). Physica Scripta. 1082. Retrieved from https://scholarworks.umass.edu/math_faculty_pubs/1082

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NLS in damping linear of effects the on spro- is 1 Dissipation ..Frantzeskakis D.J. , (1) e 1 where spsil,w hscnie h apdNSequation NLS general damped as form: the formulation the consider of the thus keep we to possible, as trying but [11], with Ref. equation NLS the pation of context the non- of formation waves. the water asymptotic linear of and complete of instability the the context of to the suppression lead in may [9,10] dissipation argued waves, it recently as been particular, the in has In interest special models. of dissipative property above-mentioned settings a solitary above is the of MI of formation all [1–5,8,9,17,18]. the in to structures leads coherent MI and waves solu- of (MI) onset (CW), continuous-wave The instability alias tions. modulational uniform, the spatially is of equation NLS the 5. ab- photons to the resonant close of the is frequency of carrier absorption ratio the the the to of frequency that sorption fact order nonlin- the high Kerr by the explained the case, is this with In waveguide silica [16]. earity planar a o in self-focusing in light collapse mecha- spatiotemporal a takes arresting as that nism scattering) term Raman stimulated cubic the a regard into with (along role important cially for trains soliton matter-wave of evolution [15]. scattering and negative with [14] BECs length in re- in collapse introduced of also studies of were cent forms study terms complex to Such more necessary with terms. dissipative type is NLS/GP it the Fesh- of Thus, models the [12,13]). crossing resonance as bach (such losses dynamically-induced eladpstv i re o h orsodn emto term corresponding the for order (in positive and real eig e vvUiest,TlAi 97,Israel 69978, Aviv Tel University, Aviv Tel eering, h i ftepeetwr st xmn h Iin MI the examine to is work present the of aim The of features dynamical important most the of One accounting term a of form the in – loss Higher-order five-photon t a aepae n corresponding a and place, take may wth oiae ytemdlfrBC eeoe in developed BECs for model the by Motivated . to.Tesuyi oiae yresults by motivated is study The ation. scuet,AhrtM 10-55 USA 01003-4515, MA Amherst ssachusetts, b clrslsaecrooae ynumer- by corroborated are results ical iu e ntennierSchrödinger (NLS) nonlinear the in tes odnae BC) swl sb the by as well as (BECs), condensates oi,Zgao,Ahn 58,Greece 15784, Athens Zografos, polis, and t pto om nldn h constant the including form, ipation pessteisaiiyo longer a on instability the ppresses 2 = n ..Malomed B.A. and c − 1 bopin–wsas on opa cru- a play to found also was – absorption u r elprmtr.Teconstant The parameters. real are xx + b | u | 2 e qainwith Equation ger u ( + c 3 1 − ic 2 ) | u olna dissi- nonlinear | 2 k u, c 2 (2) is f describe loss, rather than gain). The value k = 0 brings f(t) ≡−kc2/F (t), (8) one back to the much studied case of the linear dissipa- 2 2 g(t) ≡−4k c2/F (t) tion [1,7–9], while k = 2 corresponds to the quintic loss, 2 2 1/k accounting for the above-mentioned three-body recombi- + q q +2b/ (F (t)) +2kc1/F (t) . (9) nation collision in BECs [11], and k = 5 – to the above- h i mentioned case of the five-photon absorption, which is Equation (7) can be further cast in a more standard believed to be the mechanism arresting the spatiotempo- form,y ¨ + h(t)y = 0, where h(t) ≡ g(t) + f 2(t), and ral collapse in glass [16]. t y ≡ wr(t)exp f(s)ds . Our presentation is structured as follows. In Section 0 R II, we consider the MI in the general equation (2). In It follows from Eq. (9) that, in the case of the self- Section III, we focus on three special cases: (i) the (pre- focusing nonlinearity (b, c1 < 0), the initial condition w˙ (0) = 0 (which we will adopt below in numerical simu- viously studied) linear damping case, (ii) c2 = 0 [the r Hamiltonian limit of Eq. (2)], and (iii) the case of purely lations) leads to g(0) < 0, provided that the wavenumber is chosen from the interval dissipative nonlinearity, with c1 = b = 0. In Section IV, we complement our analytical findings by numerical 2 2 2 2k simulations. Section V concludes the paper. q < qcr ≡−(bB0 + kc1B0 ) 2 2k 2 2 2 4k + (bB0 + kc1B0 ) +4k c2B0 . (10) q II. MODULATIONAL INSTABILITY FOR THE DISSIPATIVE NLS MODEL The condition g(0) < 0 implies an initial growth of the perturbation, similar to the case of MI in the usual NLS equation (1) [1,2,18]. It is interesting to note that, some- An exact CW (spatially uniform) solution to Eq. (2) what counter-intuitively, the interval (10) of unstable is 2 wavenumbers expands (qcr increases) with the increase 1 −1/(2k) ib 1− k of the loss constant c2. This result will be verified by u0(t) = (F (t)) exp (F (t)) 2c2(1 − k) means of numerical simulations in Sec. IV. However, the perturbation growth can only be tran- i c1 × exp − ln F (t) − β0 , (3) sient in the presence of the loss, which resembles a known 2c2 k situation for the MI in a conservative medium where the where β0 is an arbitrary phase shift, dispersion coefficient changes its sign in the course of the evolution (this actually pertains to evolution of an −2k F (t)= B0 +2kc2t, (4) optical signal along the propagation distance in an inho- mogeneous dispersive medium) [19]. In particular, even and B0 is a positive constant. We now look for perturbed when g(0) is negative (i.e., when the condition (10) is sat- solutions of the NLS equation in the form isfied and we are in the –transiently– unstable regime), as t increases, g(t) increases monotonically as well and u(x, t)= u0(t)[1+ ǫw(t)cos(qx)] , (5) eventually becomes positive. Thus, we may expect that, until a critical time t0 > 0, the function wr monotoni- where w(t) and q are the amplitude and constant cally grows but then switches from growth to oscillations. wavenumber of the perturbation, while ǫ → 0 is an This prediction also follows from the consideration of the infinitesimal perturbation parameter. Substituting (5) asymptotic form of Eqs. (8) and (9) for t → ∞: as the into (2), keeping terms linear in ǫ, and splitting the dissipation suppresses f(t) and the time-dependent part perturbation amplitude in its real and imaginary parts, of g(t), in this limit Eq. (7) gives rise to oscillations of w ≡ w + iw , we obtain the following ordinary differen- 2 r i wr at the frequency q (and, hence, to stable dynamics). tial equations (ODEs)

2 −1 w˙ r = q wi +2kc2 (F (t)) wr, III. SPECIAL CASES 2 −1/k −1 w˙ i = − q +2b F (t) +2kc1 (F (t)) wr, (6) where the overdot stands for d/dt. These equations can In this section we consider, in more detail, particu- be combined into the following linear second-order equa- lar cases of special interest: (i) k=0 (ii) c2 = 0, and tion for wr, (iii) b = c1 = 0. The ﬁrst case is the well-known linear damping limit (given for completeness). The second case w¨r +2f(t)w ˙ r + g(t)wr =0, (7) corresponds to a higher-order version of the conservative NLS equation [see Eq. (1)], while the last one is a case where of a purely dissipative nonlinearity.

2 2 4 A. Linear Dissipation 2 d wr dwr q 2 τ 2 + τ + 2 2 τ − 1 wr =0, (16) dτ dτ 4c2k In the limiting case k → 0, Eqs. (2)-(7) lead to the fol- which is the Bessel equation. Its solutions are lowing results: first, the relevant exponentially decaying solution to (2) reads q2τ q2τ wr(τ)= A1J1 + A2Y1 , (17) 2c2k 2c2k b u0(t) = exp(−c2t)exp i exp(−2c2t) − c1t + β0 . 2c2 where A1,2 are arbitrary constants, and J1, Y1 are the Bessel functions of the first and second kind, respec- (11) tively. According to the known asymptotic properties of the Bessel functions, this solution clearly shows tran- On the other hand, w now satisfies the much simpler r sition to an oscillatory behavior for large τ. Notice that equation the case of b = 0 and c1 6= 0 is explicitly solvable too, ′′ 2 2 through hypergeometric and Laguerre functions, but the w + q q +2b exp(−2c2t) wr =0, (12) r final expressions are rather cumbersome (and hence not which can be identified, e.g., with the equation (15.2.1) shown here). of [1] (see also [7,8]). An explicit, though complicated, solution to this equation can be found using Bessel func- IV. NUMERICAL RESULTS tions [8] and the critical time can be calculated explicitly (see e.g., [1]). The above results indicate that the linearized equation governing the evolution of the modulational perturba- B. The conservative model. tions can be explicitly solved only in special cases. In order to analyze the general case, we verified the long- time asymptotic predictions following from Eq. (7) by In the case c = 0, an exact CW solution to Eq. (2) is 2 numerical simulations of this equation in the case which 2 2k is most relevant to the BECs. As mentioned above, this u0(t)= A0 exp −i(bA0 + c1A0 )t , (13) one corresponds to k = 2, while the other parameters in where A0 is an arbitrary constant, and the linearized Eq. (2) are set, without loss of generality, to b = −1 and equation (7) assumes the form c1 = 0 (the latter condition is adopted since, typically in the applications of interest, the conservative part of the ′′ 2 2 2 2k wr + q q + 2(bA0 + kc1A0 ) wr =0, (14) quintic term is less significant than the dominant cubic term). Lastly, we fix B0 = 1 in the CW solution (4), 2 2 thus giving rise to the MI criterion, q < −2(bA0 + which is also tantamount to the general case, through 2k kc1A0 ). As it follows from here, MI is possible if ei- a scaling transformation of Eq. (2), provided that c2 ther nonlinearity is self-focusing, i.e., either b < 0 or is kept as a free parameter. Equation (7) was solved us- c1 < 0. If b < 0 and the higher-order nonlinear terms ing a fourth-order Runge-Kutta integrator, with the time are also focusing, then the interval of modulationally un- step dt = 0.001, and initial conditions wr(0) = 0.05 and stable wavenumbers grows. On the other hand, if the w˙ r(0) = 0. higher order terms are defocusing , then the correspond- Figure 1 illustrates the critical wavenumber bounding ing interval shrinks and does so faster for higher powers the MI band (of transient instability, if c2 6= 0) as a (or BEC densities). In particular, the higher-order self- function of the dissipation strength c2. The solid line defocusing nonlinearity (k > 1 and c1 > 0) completely is the analytical prediction given by Eq. (10), which is suppresses the MI if the CW amplitude is large enough, compared with points representing numerical data. The 2(k−1) A0 > |b|/ (kc1). latter were obtained by identifying the wavenumber for each value of c2, beyond which there was no initial growth of the perturbation wr. Good agreement is between the C. The model with dissipative nonlinearity theoretical prediction and the numerical observations is obvious. Figure 2 shows the difference in the evolution of mod- In the case of b = c1 = 0, Eq. (7) takes a simple form, ulational perturbations whose wavenumber is chosen in- ′′ ′ 4 2 2 2 side (left panels) or outside (right panels) the instability wr + 2 [kc2/F (t)] wr + q − 4c2k / (F (t)) wr =0. h i band, in the absence (top panels) and in the presence (15) (middle and bottom panels) of the quintic dissipation. The top left panel shows the case of q = 1.4 for c2 = 0, Upon setting τ ≡ F (t), Eq. (15) reduces to the top right illustrates the case of q =1.5 for the same

3 c2, while the middle panels demonstrate the evolution of Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. wr for |c2| = 0.3, for q = 1.4 (left panel) and q = 1.7 Phys. 71, 463(1999). (right panel). The bottom panels show the evolution [7] D. Anderson and M. Lisak, Opt. Lett. 9, 468-470 (1984). of the norm in the GP equation (right panel) and the [8] M. Karlsson, J. Opt. Soc. Am. B 12, 2071 (1995). spatio-temporal evolution of a function representing the [9] H. Segur, D. M. Henderson, J.L. Hammack, C.-M. Li, Stabilizing the Benjamin-Feir perturbation, illustrating (for q = 1) the initial growth D. Pheiff and K. Socha, Instability, submitted to J. Fluid Mech. (2004). and eventual stabilization. [10] H. Segur, Stabilizing the Benjamin-Feir Instability, In- vited talk at the Fields Institute, available at: http://www.fields.utoronto.ca/audio/03- V. CONCLUSIONS 04/physics patterns/segur/ (2003). [11] T. Köhler, Phys. Rev. Lett. 89, 210404 (2002). In this work we studied the modulational instability [12] E. A. Donley et al., Nature 412, 295 (2001). J. L. Roberts (MI) in equations of the NLS type in the presence of et al., Phys. Rev. Lett. 81, 5109 (1998); S. L. Cornish et nonlinear dissipative perturbations, motivated by the real., Phys. Rev. Lett. 85, 1795 (2000); [13] S. Inouye et al., Nature 392, 151 (1998); J. Stenger et cent developments in the fields of nonlinear optics, wa- al., Phys. Rev. Lett. 82, 2422 (1999). ter waves and Bose-Einstein condensates (BECs). For [14] Yu. Kagan, A. E. Muryshev, and G. V. Shlyapnikov, general power-law nonlinearities with both conservative Phys. Rev. Lett. 81, 933 (1998); H. Saito and M. Ueda, and dissipative parts, we analyzed the spatially uniform Phys. Rev. Lett. 86, 1406 (2001). (CW) solutions and their stability. We found that the [15] V. Y. F. Leung, A. G. Truscott, and K. G. H. Baldwin, dissipation generically suppresses the MI, switching the Phys. Rev. A 66, 061602 (2002). growth of the perturbations into an oscillatory behavior. [16] H. Eisenberg et al., Phys. Rev. Lett. 87, 043902 (2001). We identified some special cases that can be analyzed [17] C. Sulem and P. L. Sulem, The Nonlinear Schrödinger exactly. Analytical predictions based on the asymptotic Equation, Springer-Verlag (New York, 1999). consideration for t → ∞ were compared with numeri- [18] A. Smerzi et al., Phys. Rev. Lett. 89, 170402; L. Salas- cal simulations. A noteworthy prediction (confirmed by nich, A. Parola, and L. Reatto Phys. Rev. Lett. 91, 080405 (2003); G. Theocharis et al., Phys. Rev. A 67, numerical results) is that the band of wavenumbers for 063610 (2003). expands which the transient MI occurs with the increase [19] B. A. Malomed, Physica Scripta 47, 311 (1993). of the strength of the dissipation. [20] K. E. Strecker et al., Nature 417, 150 (2002); F. S. Catal- As the MI is currently amenable to experimental ob- iotti et al., New J. Phys. 5, 71 (2003). servation in BECs [20], it would be particularly interesting to conduct experiments with different densities of the condensate. With a higher density, three-body recombination effects and the corresponding nonlinear dissipation are expected to be stronger, affecting the onset of the MI as predicted in this work. PGK gratefully acknowledges support from NSF-DMS- 0204585, NSF-CAREER and from the Eppley Founda- tion for Research. Harvey Segur is gratefully acknowl- edged for valuable discussions and for providing us a 2.4 preprint of Ref. [9]. 2.2

2 cr q

1.8

[1] A. Hasegawa and Y. Kodama, Solitons in optical Com- munications, Clarendon Press (Oxford 1995) 1.6 [2] T. B. Benjamin and J. E. Feir, J. Fluid. Mech. 27, 417

(1967). 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 c [3] L. Bergé, Phys. Rep. 303, 260 (1998). 2 [4] P. K. Shukla, M. A. Hellberg, and L. Stenflo, J. Atmos. FIG. 1. The critical wavenumber qcr [see Eq. (10)] of the Solar-Terr. Phys. 65, 355 (2003). band which gives rise to the initial growth of modulational [5] T. Farid, P. K. Shukla, A. M. Mirza, and L. Stenflo, Phys. perturbations around the CW solution. The solid line is the Plasmas 7, 4446 (2000). analytical prediction (10), and the points show the numeri- [6] L. P. Pitaevskii and S. Stringari, Bose Einstein Conden- cal results. The parameters are B0 = 1, b = −1, k = 2 and sation, Clarendon Press (Oxford, 2003); F. Dalfovo, S. c1 = 0.

4 120 0.05 FIG. 2. Top panels display the evolution of the modula- 0.04 100 tional perturbation, wr, in the absence of dissipation (c2 = 0), 0.03

0.02 80 for wavenumbers that do (q = 1.4, left panel) or do not 0.01 (q = 1.5, right panel) belong to the instability band. The (t) (t)

r 60

r 0 w w

−0.01 middle panels display the same, but in the presence of the

40 −0.02 nonlinear quintic dissipation (|c2| = 0.3), for the perturba-

−0.03 20 tion wavenumbers q = 1.4 (left panel) and q = 1.7 (right −0.04

0 −0.05 panel). The bottom panels show a typical result from the full 0 5 10 15 20 25 30 0 5 10 15 20 25 30 t t 0.06 0.05 simulation of the Gross-Pitaevskii equation for q = 1. The 0.04 left panel shows the space-time evolution of (|u| − |u0|)/|u0|, 0.04 0.03 which evolves in a periodic form, whose amplitude depends 0.02 0.02 0.01 on w. We can see in the colorbar, the initial transient growth, (t) (t)

r 0 r 0 w w followed by the dissipative evolution. The right panel shows −0.01

−0.02 −0.02 the best ﬁt (solid line) and numerical (dashed line practi-

−0.03 −0.04 cally coinciding with the solid one) time evolution of the norm −0.04 P = |u|2dx. The best ﬁt exponent of log(P ) vs. log(1+1.2t) −0.06 −0.05 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t t was foundR to be −0.4994 compared with the theoretical ex- −50

0.06 ponent of −0.5 (according to Eqs. (3)-(4)). −40

−30 0.04

−20 0.02 −10

0 P

x 0

2 10 10 −0.02

20 −0.04

−0.06 40

50 −0.08 0 1 0 5 10 15 20 25 30 35 40 10 10 t t

5 −50

0.06 −40

−30 0.04

−20 0.02 −10

x 0

10 −0.02

20 −0.04

−0.06 40

50 −0.08 0 5 10 15 20 25 30 35 40 t