Nonlinear Schrödinger equation in cylindrical coordinates

Abstract

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 ${\Delta }_{\perp }={\partial }_x^2+{\partial }_y^2$ 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, ${\Delta }_{\perp }={\partial }_r^2+\frac{1}{r}{\partial }_r$ 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.

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