Vibrations and Oscillatory Instabilities of Gap Solitons

Abstract

Stability of optical gap solitons is analyzed within a coupled-mode theory. Lower intensity solitons are shown to always possess a vibration mode responsible for their long-lived oscillations. As the intensity of the soliton is increased, the vibration mode falls into resonance with two branches of the long-wavelength radiation producing a cascade of oscillatory instabilities of higher intensity solitons.[S0031-9007 (98)06265-6] PACS numbers: 42.65.Tg, 03.40.Kf, 05.45.+b, 75.30.Ds In the late 1970s and early 1980s, the theory of elementary particles [1] and condensed matter physics (in particular, the Su-Schrieffer-Heeger polyacetylene model [2]) stimulated a wide interest in particlelike solutions of classical spinor field equations. Recently there has been a remarkable upsurge of the interest; the localized solutions of spinorlike systems have made a comeback under the new name of gap solitons.Thanks to the gap in the linear spectrum, solitons in spinorlike systems can propagate without losing their energy to resonantly excited radiation waves [3,4]. An example of the gap-soliton bearing system is given by an optical fiber with periodically varying refractive index [3]; here the gap is produced by the Bragg reflection and resonance of the waves along the grating. Another class of gap solitons arises in two-wave resonant optical materials with a x ͑2͒ susceptibility and diatomic crystal lattices (see [4] and references therein). Finally, in the already mentioned polyacetylene model [2], the gap in the electron spectrum is due to the electron-phonon interaction and effective period doubling of the lattice.The aim of this Letter is to analyze the stability of gap solitons. Previous analytical studies of the spinor soliton stability faced serious obstacles (cf.[5]), while results of computer simulations were contradictory (cf. [6,7]). As a result, no stability or instability criterion is available to date. The main difficulty of the previous analyses was that they were all based on a postulate that stable solutions must render the energy minimum. In the actual fact, however, the minimality of energy is not necessary for stability in systems with indefinite metrics [8]. As far as optical gap solitons are concerned, they have been commonly deemed stable following recent computer simulations carried out for certain particular parameter values [9]. In this Letter we demonstrate that the gap solitons can be unstable, elucidate the mechanism of instability, and demarcate the stability/instability regions on the plane of their parameters.In nonlinear optics the gap solitons are usually analyzed within the coupled-mode theory [3] which reduces to a system of coupled equations for the amplitudes of the forward-and backward-propagating waves, i͑u t 1 u x ͒ 1 y 1 ͑jyj 2 1 rjuj 2 ͒u 0 , i͑y t 2 y x ͒ 1 u 1 ͑juj 2 1 rjyj 2 ͒y 0 .In the periodic Kerr medium one typically has r 1͞2 [3]; in other problems of the fiber optics r may range up to infinity [10]. In the case r 0 Eqs. (1) yield the massive Thirring model of the field the...

Approximate analytic solutions to coupled nonlinear Dirac equations

“…In particular, we have given the first (approximate) analytic solitary wave solution to two coupled NLDEs for both scalar-scalar interactions and vector-vector interactions. These solutions are relevant in nonlinear optics [5] as well as for light solitons in waveguide arrays [6][7][8] among other applications in BECs and cosmology. Further,we have shown using the Moore's decoupling method that in the nonrelativisticlimit, NLDEs with both scalar-scalar and vector-vector interactions reduce tothe same coupled nonlinear Schrödinger equation (NLSE).…”

“…The nonlinear Dirac (NLD) equation in 1 + 1 dimensions [1] has a long history and has emerged as a useful model in many physical systems such as extended particles [2][3][4], the gap solitons in nonlinear optics [5], light solitons in waveguide arrays and experimental realization of an optical analog for relativistic quantum mechanics [6][7][8], BoseEinstein condensates in honeycomb optical lattices [9], phenomenological models of quantum chromodynamics [10], as well as matter influencing the evolution of the universe in cosmology [11]. Further, the multi-component BEC order parameter has an exact spinor structure and serves as the bosonic analog to the relativistic electrons in graphene.…”

Solitons in Bragg gratings with saturable nonlinearities

“…(8). On the other hand, in the standard BG model, with the cubic nonlinearity, an approximate analytical consideration [6] and accurate numerical analysis [7,8] of Eqs. (1) reveal that only the solitons from the upper-half bandgap, 0 Յ Ͻ , are stable, while the remaining half, − Ͻ Ͻ 0, is occupied by unstable solitons (as mentioned above).…”

“…In particular, it is well known that, while all soliton solutions in the BG model with the cubic nonlinearity, based on Eqs. (1), meet the condition of dE /d Ͻ 0, only half of them are truly stable [6][7][8].…”

“…The stability of these solutions was first studied by means of the variational approximation [6], and then with the help of accurate numerical methods [7,8]. It was found that approximately half of the gap-soliton family is stable, and the other half unstable (the solitons with positive and negative intrinsic frequencies are, respectively, stable and unstable).…”