Semiclassical quantization of the one- and two-kink dark solitons

Abstract

Drawing from an analogy with a linear particle-chain model, we develop a picture of the dark optical soliton as a composite of a background and notch each enjoying a certain degree of independence. The semiclassical quantization procedure of Dashen et al. [Phys. Rev. D 11, 3424 (1975)] is modified to allow for the separate quantization of these two entities. We apply our results to the one-kink and two-kink dark solitons. For both we find that the fluctuations about the notch can be understood as a bound state of bosons held by an attractive delta potential while the background fluctuations are seen as an ensemble of oscillators with a slightly repulsive interaction. We show that the collision effects are small for the two-kink soliton. The quantum numbers emerging from this analysis are interpreted in terms of the number of particles. Moreover, the topological character of the dark soliton also appears naturally in our description. We speculate on the possibility of fractional quantum numbers and derive the repulsive interaction of two dark solitons from the framework developed for the two-kink soliton.