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On the Modulational Instability of the Nonlinear Schrödinger Equation with Dissipation Z Rapti University of Massachusetts Amherst ScholarWorks@UMass Amherst Mathematics and Statistics Department Faculty Mathematics and Statistics Publication Series 2004 On the Modulational Instability of the Nonlinear Schrödinger Equation with Dissipation Z Rapti PG Kevrekidis University of Massachusetts - Amherst, [email protected] Follow this and additional works at: https://scholarworks.umass.edu/math_faculty_pubs Part of the Physical Sciences and Mathematics Commons 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 This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. On the Modulational Instability of the Nonlinear Schr¨odinger Equation with Dissipation Z. Rapti1, P.G. Kevrekidis1, D.J. Frantzeskakis2 and B.A. Malomed3 1 Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA 2 Department of Physics, University of Athens,Panepistimiopolis, Zografos, Athens 15784, Greece 3 Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel The modulational instability (MI) of spatially uniform states in the nonlinear Schr¨odinger (NLS) equation is examined in the presence of higher-order dissipation. The study is motivated by results on the effects of three-body recombination in Bose-Einstein condensates (BECs), as well as by the important recent work of Segur et al. on the effects of linear damping in NLS settings. We show how the presence of even the weakest possible dissipation suppresses the instability on a longer time scale. However, on a shorter scale, the instability growth may take place, and a corresponding generalization of the MI criterion is developed. The analytical results are corroborated by numer- ical simulations. The method is valid for any power-law dissipation form, including the constant dissipation as a special case. I. INTRODUCTION dynamically-induced losses (such as crossing the Fesh- bach resonance [12,13]). Thus, it is necessary to study models of the NLS/GP type with more complex forms of The nonlinear Schr¨odinger (NLS) equation is well dissipative terms. Such terms were also introduced in re- known to be a generic model to describe evolution of cent studies of collapse in BECs with negative scattering envelope waves in nonlinear dispersive media. This equa- length [14] and evolution of matter-wave soliton trains tion applies to electromagnetic fields in optical fibers [1], [15]. deep water waves [2], Langmuir waves [3] and other types Higher-order loss – in the form of a term accounting of perturbations in plasmas, such as electron-acoustic [4] for five-photon absorption – was also found to play a cru- waves and waves in dusty plasmas [5], macroscopic wave cially important role (along with a cubic term that takes functions of Bose-Einstein condensates (BECs) in dilute into regard the stimulated Raman scattering) as a mecha- alkali vapors [6] [in the latter case, it is usually called the nism arresting spatiotemporal collapse in self-focusing of Gross-Pitaevskii (GP) equation], and many other physi- light in a planar silica waveguide with the Kerr nonlin- cal systems. The NLS typically appears in the Hamilto- earity [16]. In this case, the high order of the absorption nian form is explained by the fact that the ratio of the resonant ab- sorption frequency to the carrier frequency of the photons iu = −u + b|u|2u, (1) t xx is close to 5. One of the most important dynamical features of where u is a complex envelope field and b represents the the NLS equation is the modulational instability (MI) strength of the nonlinearity (for instance, in the BEC of spatially uniform, alias continuous-wave (CW), solu- context, u is the mean-field wave function and b is pro- tions. The onset of MI leads to the formation of solitary portional to the s-wave scattering length [6]). The cases waves and coherent structures in all of the above settings b > 0 and b < 0 correspond, respectively, to the self- [1–5,8,9,17,18]. MI is a property of special interest in the focusing and self-defocusing NLS equation. above-mentioned dissipative models. In particular, as it In many physically relevant cases, dissipation, which has been recently argued [9,10] in the context of water is ignored in Eq. (1), is not negligible. In particular, the waves, dissipation may lead to the complete asymptotic fiber loss gives rise to a linear dissipative term in the NLS suppression of the instability and of the formation of non- equation for optical fibers [1,7,8]. Dissipative phenomena linear waves. have also been studied recently in the context of two- arXiv:cond-mat/0404597v1 [cond-mat.soft] 24 Apr 2004 The aim of the present work is to examine the MI in dimensional water waves [9,10] in order to explain exper- the context of the NLS equation with nonlinear dissi- imental observations in water tanks. In this setting also, pation. Motivated by the model for BECs developed in the dissipation is accounted for by a linear term. In fact, Ref. [11], but trying to keep the formulation as general it was the viewpoint of [9,10] on the effects of damping as possible, we thus consider the damped NLS equation that chiefly motivated the present work. Furthermore, of the form: it has been argued that dissipative phenomena are quite 2 2k relevant in the context of BECs [11,12]. However, in the iut = −uxx + b|u| u + (c1 − ic2)|u| u, (2) latter case, the loss arises from higher-order processes (such as three-body recombination, corresponding to a where b and c1 are real parameters. The constant c2 is quintic dissipation term in the GP equation [11]) or from real and positive (in order for the corresponding term to 1 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.
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