Dynamics of Dark Solitons

-Spatial dark solitons are known to be stable nonlinear localized waves in self-defocusing media. We demonstrate how to analyse the dynamics of dark solitons in the presence of small perturbations, and we consider the effect of two-photon absorption on dark solitons as a particular example. We also predict a new type of dark solitons of circular symmetry, ring dark solitons, which are stable against transverse perturbations but slowly change their parameters. Ring dark solitons may coexist with dark strips or optical vortex solitons displaying almost elastic interactions. 1. I N T R O D U C T I O N Light solitons in time (temporal solitons) and space (spatial solitons) have been the object of intensive theoretical and experimental studies during the last three decades. The solitons, localized-in-time optical pulses or bounded-in-space optical beams, evolve from nonlinear change in the refractive index of the material, induced by the light-intensity distribution. When the combined effects of the refractive index nonlinearity and the pulse dispersion (in the case of temporal solitons) or diffraction (in the case of spatial solitons) exactly compensate each other, the pulse (or beam) propagates without change of its shape, being self-trapped by the waveguide nonlinearity. The nonlinear effects which are responsible for the soliton formation are, in general, Kerr-like effects, inducing local index changes proportional to the local light power. In this case the main nonlinear equation governing the pulse/beam evolution is the famous nonlinear Schr6dinger (NLS) equation. In the case of the anomalous group-velocity dispersion (GVD) in fibres or the self-focusing nonlinearity in planar waveguides, the continuous-wave (cw) solution of the NLS equation becomes modulationally unstable and breaks into a chain of localized pulses, the so-called bright solitons. Soliton propagation of bright optical solitons has been verified in a number of elegant experiements (see, e.g. a pioneer work by Mollenauer et al. [1]). In the case of the normal GVD in fibres or the self-defocusing nonlinearity in waveguides, there are no bright solitons, instead pulses (or beams) undergo enhanced dispersive (diffractive) broadening and chirping. However, in this case the cw solution is modulationally stable, and the soliton pulses appear as localized nonlinear excitations of a background wave. The interest in analyzing dark soliton propagation in optical models has been initiated by several experimental observations of temporal dark solitons in optical fibres [2-4] and spatial dark solitons [5-7]. We note here that such an experimental success