Vibration modes of a gap soliton in a nonlinear optical medium

We analyze the dynamics of small internal vibrations in a two-component gap soliton. The general model considered describes at least three different nonlinear optical systems: a pair of waves coupled by the Bragg scattering in a medium with a periodic grating, a twisted birefringent fiber, and a dual-core asymmetric coupler. In all the cases the material dispersion of the medium is neglected, but an effective dispersion is induced by the linear coupling between the two modes. Employing the averaged Lagrangian variational technique we derive a system of ordinary differential equations which approximates the dynamics of the gap soliton. We find three oscillation modes, which are composed of (mixtures of) dilation-contraction of each component's width, and a relative translation of the two components. At certain values of the parameters the analysis yields spurious instabilities, which is a novel failure of the averaged Lagrangian variational technique.