三次-五次非线性薛定谔方程中亮涡的存在性、稳定性和动力学
Existence, Stability, and Dynamics of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger Equation
摘要：
我们重新讨论了二维五阶非线性Schr{<$}dinger方程中孤立涡（又称涡孤子）的存在性和方位调制稳定性.我们发展了一种半解析方法，假设涡旋孤子是相对窄的，从而分裂成径向和方位角一维方程的完整的二维方程。变分法是用来预测的涡旋孤子的径向形状，使用径向方程，产生的结果非常接近从数值解得到的。通过这种方法，以前已知的存在边界的孤立涡恢复。利用一维方位运动方程分析了涡旋孤子的调制不稳定性。半解析预测-特别是，在不稳定边界的涡旋孤子的临界固有频率-进行了比较，系统的二维模拟。我们还将我们的研究结果与早期作品中报道的结果进行了比较，这些结果存在一些差异。然后，我们进行了详细的计算研究与不同的拓扑电荷的稳定涡旋之间的碰撞。确定了弹性碰撞和破坏性碰撞之间的边界。
英文摘要：
We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex solitons) in the two-dimensional (2D) cubic-quintic nonlinear Schr{ö}dinger equation. We develop a semi-analytical approach, assuming that the vortex soliton is relatively narrow, and thus splitting the full 2D equation into radial and azimuthal 1D equations. A variational approach is used to predict the radial shape of the vortex soliton, using the radial equation, yielding results very close to those obtained from numerical solutions. Previously known existence bounds for the solitary vortices are recovered by means of this approach. The 1D azimuthal equation of motion is used to analyze the modulational instability of the vortex solitons. The semi-analytical predictions -- in particular, that for the critical intrinsic frequency of the vortex soliton at the instability border -- are compared to systematic 2D simulations. We also compare our findings to those reported in earlier works, which featured some discrepancies. We then perform a detailed computational study of collisions between stable vortices with different topological charges. Borders between elastic and destructive collisions are identified.
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