Nonlinear Schrödinger equation in cylindrical coordinates

Nonlinear Schrödinger equation was originally derived in nonlinear optics as a model for beam propagation, which naturally requires its application in cylindrical coordinates. However, the derivation was done in the Cartesian coordinates with the Laplacian ${\mathrm{\Delta }}_{\perp }={\mathrm{\partial }}_x^2+{\mathrm{\partial }}_y^2$ transverse to the beam $z$-direction tacitly assumed to be covariant. As we show, first, with a simple example and, next, with a systematic derivation in cylindrical coordinates, ${\mathrm{\Delta }}_{\perp }={\mathrm{\partial }}_r^2+\frac{1}{r}{\mathrm{\partial }}_r$ must be amended with a potential $V(r)=-\frac{1}{r^2}$, which leads to a Gross-Pitaevskii equation instead. Hence, the beam dynamics and collapse must be revisited.