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 ⊥ = ∂ x 2 + ∂ y 2 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 ⊥ = ∂ r 2 + 1 r ∂ r must be amended with a potential V ( r ) = 1 r 2 , which leads to a Gross–Pitaevskii equation instead. Hence, results for beam dynamics and collapse must be revisited in this setting.