Perturbation theory for gap solitons

The system of coupled nonlinear evolution equations describing gap solitons is shown to be approximately equivalent in the neighborhood of the soliton solution to the massive Thirring model. The equivalence is then used for construction of the perturbation theory for gap solitons based on the RiemannHilbert problem. The value of deviation of the initial condition from the gap soliton shape serves as the small parameter. The perturbation theory is valid for the arbitrary value of the self-phase modulation term. 03.40KfWaves and wave propagation: general mathematical aspects Typeset using REVTEX On leave from: DOPIT, Nat. Acad. Sc. of Belarus, 220141 Minsk, Zhodinskaya St. 1/2, Belarus. 1 I. RELATION BETWEEN GAP AND MTM SOLITONS Gap solitons in nonlinear optical fibers with grating have recently attracted a great deal of attention (see, e. g., [1–6] and references therein). The term gap solitons refers to the solitary waves which have their power spectrum lying well within the frequency band gap of the periodic grating applied to the optical fiber. The gap soliton solution was first obtained, in a particular form, in [1] and called there the slow Bragg soliton. The general family of gap solitons was then derived in [2]. There it was shown that gap solitons partially share the properties of the massive Thirring model (MTM) solitons, in particular, they exhibit stability under collisions. In dimensionless units, gap solitons are described by the following system of nonlinear equations i (E1T −E1X) + E2 + ( |E2| + ρ|E1| ) E1 = 0, (1.1) i (E2T + E2X) + E1 + ( |E1| + ρ|E2| ) E2 = 0, where E1 and E2 are the slowly varying envelopes of two counterpropagating waves coupled through the Bragg scattering induced by the grating (represented through the linear crosscoupling terms), the nonlinear terms contain the selfand cross-phase modulation effects. The parameter ρ at the self-phase modulation (SPM) term may range up to infinity [7], in which case the system (1.1) describes the nonlinear dual-core asymmetric coupler [4]. In the case ρ = 0 (1.1) reduces to the MTM of the field theory [8], which is integrable by the inverse scattering transform (IST) method [9–12]. The MTM is invariant with respect to the Lorentz transformations. Although the SPM term breaks the Lorentz invariance, nevertheless, the system (1.1) possesses one-soliton solution which, written in terms of the boosted variables, is, up to scalar multipliers and a nontrivial phase, one-soliton solution of the MTM [2]: