Existence Of Vortices For The Two Nonlinear Optical Models

The formation of optical vortices as dark holes is an area of extensive research in modern optical physics. In this paper, we study two nonlinear optical vortices models.The first model is two incoherently-coupled laser beams in a photorefractive crystal, where governing equations are Schrodinger equations. Some existence theorems are established for steady-state solutions of nonlinear Schrodinger equations via the variational method and the mountain-pass-theorem. According to the constrained minimization approach, we prove the existence of positive radially symmetric solutions and give some lower estimate of wave propagation constant ?. Moreover, the existence theorem of saddle-point solutions is established by mountain-pass-theorem. The second model is nonlinear Kerr model of two coupling beams propagation. We prove that the model has no non-zero steady-state solution. Besides, we also obtain an existence theorem governing the positive radially symmetric solutions via the constrained minimization approach, and give the range of the parameter ?.