Vortex patch equilibria of the Euler equation and random normal matrices
Vortex patch equilibria of the Euler equation and random normal matrices
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DOI：
10.1088/1751-8113/47/21/212002
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发表时间：
2014-05
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影响因子：
--
通讯作者：
D. Crowdy
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文献类型：
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作者：
D. Crowdy
将理想流体欧拉方程的涡片平衡或V态平衡与随机正态矩阵模型的平面极限作了新的类比。在物理上，前者是一个完全不同于Hele-Shaw流或拉普拉斯增长的流体动力学问题，后者与矩阵模型的类比在最近几年已经广为人知。随机矩阵与涡旋动力学之间的联系是通过所谓的修正Schwarz势来实现的。这种理论联系虽然本身很有趣，但通过转移已经为涡旋动力学开发得很好的数学技术，立即对随机矩阵理论产生了影响。因此，对于平面极限中的多支点矩阵模型，我们描述了一种构造方法，即利用基本代数曲线的肖特基模型中的自同构共形映射和肖特基-克莱因素函数，求出给定势的特征值支集的形状。作为例子，给出了四次位势的两支点本征值分布。
A new analogy is drawn between vortex patch, or V-state, equilibria of the Euler equations for ideal fluids and the planar limit of random normal matrix models. Physically the former is a quite different fluid dynamical problem to Hele-Shaw flow, or Laplacian growth, to which an analogy with matrix models has become well known in recent years. The connection of random matrices with vortex dynamics is made via the so-called modified Schwarz potential. This theoretical link, while interesting in itself, has immediate ramifications for random matrix theory by virtue of a transfer of mathematical technology already well developed for vortex dynamics. Hence for multi-support matrix models in the planar limit we describe a constructive approach to the inverse problem of finding the shapes of the eigenvalue supports for a given potential using automorphic conformal mappings within a Schottky model of the underlying algebraic curves and use of the Schottky–Klein prime function. Two-support eigenvalue distributions in a quartic potential are given as an illustrative example.