Half-Skyrmions and Spike-Vortex Solutions of Two-Component Nonlinear Schrödinger Systems

Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of nonlinear Schrödinger equations (NLSE) describing a binary mixture of Bose-Einstein condensates (cf. [2] and [25]). Besides, half-skyrmions characterized by half-integer topological charges can also be found in the nonlinear σ model which is a model of the Bose-Einstein condensate of the Schwinger bosons (cf. [18]). Here we prove rigorously the existence of half-skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of NLSE on the entire plane R2. These spike-vortex solutions having spikes in one component and a vortex in the other component may form half-skyrmions. By Liapunov-Schmidt reduction process, we may find spike-vortex solutions of two-component systems of NLSE.