Multi-component vortex solutions in symmetric coupled nonlinear Schrödinger equations

A Hamiltonian system of incoherently coupled nonlinear Schrödinger (NLS) equations is considered in the context of physical experiments in photorefractive crystals and Bose-Einstein condensates. Due to the incoherent coupling, the Hamiltonian system has a group of various symmetries that include symmetries with respect to gauge transformations and polarization rotations. We show that the group of rotational symmetries generates a large family of vortex solutions that generalize scalar vortices, vortex pairs with either double or hidden charge, and coupled states between solitons and vortices. Novel families of vortices with different frequencies and vortices with different charges at the same component are constructed and their linearized stability problem is block-diagonalized for numerical analysis of unstable eigenvalues.