Nonlinear Schr\"{o}dinger equation in cylindrical coordinates

TL;DR

This paper revisits the nonlinear Schrödinger equation in cylindrical coordinates, revealing the need for an additional potential term and its implications for beam dynamics and collapse in nonlinear optics.

Contribution

It provides a systematic derivation of the nonlinear Schrödinger equation in cylindrical coordinates, highlighting the necessity of a potential term previously overlooked.

Findings

The Laplacian in cylindrical coordinates requires an additional potential term.

The corrected equation is a form of the Gross-Pitaevskii equation.

Beam dynamics and collapse behavior must be reexamined with the new formulation.

Abstract

Nonlinear Schr\"{o}dinger 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.