非线性薛定谔方程中涡旋的方位调制不稳定性
Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation
摘要：
在聚焦二维非线性薛定谔（NLS）方程中研究了具有不同拓扑荷的涡旋的方位调制不稳定性。研究稳定性的方法依赖于在NLS的拉格朗日泛函中冻结径向方向，以形成准一维方位运动方程，然后在方位模式的傅立叶空间中应用稳定性分析。我们制定预测的增长率的个别模式，并发现，旋涡是不稳定的临界方位波数以下。首先用变分方法得到一个渐近解析解，然后用它作为数值优化程序的初始条件，得到稳态涡解。NLS的直接数值模拟的稳定性分析预测证实。我们简要地介绍了如何扩展的方法，包括非局部非线性，往往稳定的解决方案。
英文摘要：
We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schr{ö}dinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize solutions.
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