Discreteness-Induced Oscillatory Instabilities of Dark Solitons

We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities of the localized modes it supports. We present the first example of an oscillatory instability of dark solitons, and analyse how it may occur for dark solitons of the discrete nonlinear Schrödinger and generalized Ablowitz-Ladik equations. PACS numbers: 03.40.Kf, 42.65.Tg, 63.20.Pw Typeset using REVTEX 1 Wave instabilities are probably the most remarkable nonlinear phenomena that may occur in nature [1]. One of the first instabilities discovered for nonlinear models was the modulational instability, which is known to be an effective physical mechanism in fluids [2] and optics [3] for break-up of continuous modes into solitary waves. Also, the solitary waves themselves may become unstable, and the analysis of their instabilities is an important problem of nonlinear physics. Instabilities are known to occur for both bright [4] and dark [5] solitary waves of different nonintegrable nonlinear models. Recently, a new type of solitary-wave instability, oscillatory instability, has been found to occur for bright Bragg gap solitons in the generalized Thirring model [6]. Such an instability is characterized by complex eigenvalues, and its scenario is associated with a resonance between the long-wavelength radiation and soliton internal modes which appear in the soliton spectrum when the model becomes nonintegrable [7]. In spite of the fact that oscillatory instabilities appear often in dissipative models [8], their manifestation in continuous Hamiltonian models is rare [9], and so far no example has been known for oscillatory instability of dark solitons. The aim of this Letter is twofold. First, we analyse what we believe to be the first examples of oscillatory instabilities of dark solitons, by considering the important cases of the discrete nonlinear Schrödinger (DNLS) and generalized Ablowitz-Ladik (AL-DNLS) models. We reveal two different scenarios for the dark-soliton oscillatory instability, which may occur due to either a resonance between radiation modes and the soliton internal mode, or a resonance between two soliton internal modes. Second, we demonstrate that even a weak inherent discreteness may drastically modify the dynamics of a nonlinear system leading to instabilities which have no analog in the continuum limit. First, we consider the well-known DNLS equation, iψ̇n + C(ψn+1 + ψn−1) + |ψn|ψn = 0, (1) where the dot stands for the derivative in time. Stationary localized solutions of Eq. (1) in the form ψn(t) = φne , where Λ = 2C cos k+(φ), may exist as dark-soliton modes with 2 the nonvanishing boundary conditions φn → ±φ(0)eikn (n → ±∞), provided the background wave is modulationally stable, i.e. for C cos k < 0 [10]. Without loss of generality, we can put the background intensity to unity, φ = 1. The structure of the dark-soliton modes of Eq. (1) has been discussed earlier [11,12]. Here, we consider the case C > 0, in which the background wave has k = π and is ‘staggered’, as shown in Figs. 1(a,b). The transformation ψn → (−1)ψn immediately yields the corresponding ‘unstaggered’ modes (k = 0) for negative C. The dark-soliton modes presented in Figs. 1(a,b) describe two types of stationary ‘black’ solitons [5] in a discrete lattice, the on-site mode (A-mode) centered with zero intensity at a lattice site, and the inter-site mode (B-mode) centered between two sites. These two modes can be uniquely followed from the continuous limit (C → ∞) to the ‘anticontinuous’ limit (C = 0). At C = 0, the A-mode takes the form φn = (...− 1,+1,−1, 0,+1,−1,+1, ...), and it describes a single ‘hole’ in a background wave with constant amplitude and a π phase shift across the hole. Similarly, the B-mode takes the form φn = (...− 1,+1,−1,−1,+1− 1, ...), and it describes the lattice oscillation mode with a π phase shift between two neighboring sites and no hole. Linear stability of the A-mode for small enough C follows from Aubry’s theorem (Theorem 9 in Ref. [13]) relating the linear stability of multi-breathers to the extrema of the effective action which is a function of the relative phases αn of the single breathers. For the DNLS equation (1) with a positive nonlinearity, local minima of the effective action correspond to stable solutions. A perturbative expression of the effective action to order C was obtained in Eq. (A10) in [14]. For small C, it is enough to consider the phase-interactions between nearest and next-nearest neighboring sites. Then, the lowest order contribution to the effective action from neighboring excited sites is ∑ n[2C|Λ| cos(αn+1−αn)−C cos(αn+1−αn)], while the contribution from the next-nearestneighbor interaction is −C2 ∑n 6=n0 cos(αn+1 − αn−1) + 2C cos(αn0+1 − αn0−1), where n0 is the site with the ‘hole’. For the A-mode, αn+1 − αn = π, αn+1 − αn−1 = 0, n 6= n0,