Computer Physics Communications 126 (2000) 22–27 www.elsevier.nl/locate/cpc Oscillatory instabilities of gap solitons: a numerical study I.V. Barashenkov a,1,2,3 , E.V. Zemlyanaya b,2,4,5 a Department of Physics, University of Crete, P.O. Box 2208, Heraklion 71003, Greece b Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, Dubna 141980, Russia Abstract We study stability of optical gap solitons, close relatives of spinor solitons of nonlinear classical field theory. The analysis of the associated linearized eigenvalue problem demonstrates the existence of a cascade of oscillatory and translational instabilities. The numerical technique involved is a combination of the Fourier transform and continuous analogue of Newton’s method.  2000 Elsevier Science B.V. All rights reserved. PACS: 03.40.Kf; 42.65.T; 42.81.Dp; 02.60.Lj Keywords: Gap solitons; Spinor solitons; Soliton stability; Eigenvalue problems; Newtonian iterative schemes 1. Introduction, motivation and formulation of the problem This contribution is devoted to a long-standing problem of the soliton theory, namely stability of spinor solitons. Spinor solitons attracted a great deal of attention in the late seventies and early eighties, primarily in connection with extended models of elementary particles [1] and field-theoretic description of polyacetylene [2]. The problem that was intensively studied [3–5] but has remained rather mysterious, is the stability of spinor solitons. Here the main obstacle to standard analyses stems from the unusual 1 On sabbatical leave from the University of Cape Town. Perma- nent address: Department of Applied Mathematics, University of Cape Town, Private Bag Rondebosch 7701, South Africa. E-mail: igor@maths.uct.ac.za. 2 Supported by FRD of South Africa and URC of the University of Cape Town. 3 E-mail: igor@selas.physics.uch.gr. 4 Supported by RFFR (grant 97-01-01040). 5 E-mail: elena@ultra.jinr.ru. properties of the energy functional associated with the Dirac equation. The Dirac energy is not bounded both from above and from below; in fact, it cannot have even conditional local minima. For this reason the standard stability approaches appealing to local and/or conditional minima of energy or energy-based Liapunov functionals cannot work for spinor solitons. Direct computer simulations did not add much clarity either, producing conflicting results [4,5]. Recently spinor and spinor-like solitons have made a dramatic come-back under the new name of gap solitons [6–8]. An example of the gap-soliton bearing system is given by an optical fiber with periodically varying refractive index, i(u t + u x ) + v + ( |v| 2 + ρ |u| 2 ) u = 0, i(v t − v x ) + u + ( |u| 2 + ρ |v| 2 ) v = 0. (1) Here u and v are the amplitudes of the counter- propagating waves, x is the coordinate along the grating, t time and ρ a parameter. In the periodic Kerr medium one typically has ρ = 1/2 [6]; in other settings ρ may range up to infinity [9]. The spectrum 0010-4655/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII:S0010-4655(99)00241-6