Existence of Vortices for Nonlinear Schr\"{o}dinger Equations

TL;DR

This paper proves the existence of vortex solutions in nonlinear Schr"odinger equations relevant to Bose-Einstein condensates and geometric optics, using variational and minimization methods with explicit bounds.

Contribution

It introduces a weighted Sobolev space framework and applies variational and constrained minimization techniques to establish vortex existence and bounds in these equations.

Findings

Existence of positive, radially symmetric vortex solutions.

Development of a weighted Sobolev space for the Gross-Pitaevskii equation.

Explicit estimates for wave propagation constants.

Abstract

In this paper, we study the existence of vortices for two kinds of nonlinear Schr\"{o}dinger equations arising from the Bose-Einstein condensates and geometric optics arguments, respectively. For the Gross-Pitaevskii equation from Bose-Einstein condensates arguments, we introduce the weighted Sobolev space on which the corresponding functional is coercive. By using the variational methods, we prove the existence of positive and radially symmetric solutions under different types of boundary condition. And we study another equation arising from geometric optics arguments by constrained minimization method. Furthermore some explicit estimates for the bound of the wave propagation constant are also derived.