The nonlinear Schrödinger equation in cylindrical geometries

Abstract

The nonlinear Schrödinger equation was originally derived in nonlinear optics as a model for beam propagation, which naturally requires its application in cylindrical coordinates. However, that derivation was performed in Cartesian coordinates for linearly polarized fields with the Laplacian Δ⊥=∂x2+∂y2 transverse to the beam z-direction, and then, tacitly assuming covariance, extended to axisymmetric cylindrical setting. As we show, first with a simple example and next with a systematic derivation in cylindrical coordinates for axisymmetric and hence radially polarized fields, Δ⊥=∂r2+1r∂r must be amended with a potential V(r)=1r2 , which leads to a Gross–Pitaevskii equation instead. Hence, results for beam dynamics and collapse must be revisited in this setting.