Existence of optical vortices in R2. (December 2019) - Guo, Qing; Cao, Daomin; Li, Hang
Abstract: Optical vortices are phase singularities nested in electromagnetic waves which constitute a fascinating source of phenomena in the physics of light and display deep similarities to their close relatives, quantized vortices in superfluids and Bose–Einstein condensates. The present paper is concerned with the existence of stationary optical vortices wave solutions of nonlinear Schrödinger equations in R 2 . For the self-focusing case, we consider three types of problems. The first type concerns the existence of positive radial solutions obtained by a constrained minimization approach. The second type addresses the existence of saddle-point solutions through a mini–max method. As to the third type, we use the variational argument and Nehari manifold to establish nodal solutions. On the other hand, for the defocusing case, we also prove the existence of solutions by constructing approximate sequences of solutions to overcome the loss of compactness. Moreover, in both cases, some non-existence results are discussed as well.