论文标题
圆柱坐标中的非线性schrödinger方程
Nonlinear Schrödinger equation in cylindrical coordinates
论文作者
论文摘要
非线性schrödinger方程最初是在非线性光学元件中作为光束传播的模型得出的，该模型自然需要其在圆柱坐标中的应用。但是，该派生是在载有拉普拉斯$δ_ {\ perp} = \ partial_ {x}^{2} + \ partial_ {y}^{2} $横向到beam $ z $ z $ -Direction tacitly的tacitly假定为COVARIANT的tAciLAINT。正如我们所显示的，首先，有一个简单的示例，接下来，在圆柱坐标中具有系统的推导，$δ_{\ perp} = \ partial_ {r}^{2} + \ frac {1} {1} {r} {r} {r} {r partial_ {r} $ $ v（r）= - \ frac {1} {r^{2}} $，它会导致gross-pitaevskii方程。因此，必须重新审视光束动态和崩溃。
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 $Δ_{\perp} = \partial_{x}^{2} + \partial_{y}^{2}$ 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, $Δ_{\perp} = \partial_{r}^{2} + \frac{1}{r} \partial_{r}$ 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.
