$N$点涡系的自相似运动及相关的相对平衡
Self-similar motions and related relative equilibria in the $N$-point vortex system
摘要：
研究了点涡系统的自相似解。对三点涡问题和四点涡、五点涡问题的一个具体例子，得到了自相似解的显式表达式。我们看到，这些自相似坍缩解构成的族是由单参数族描述的，它们的坍缩时间和哈密顿量也由相同参数的函数表示。然后，在参数的极限点处的配置处于相对平衡。对于多涡问题，我们用数值计算的方法研究了点涡系统。特别地，考虑到$N - 1$点涡具有均匀涡强度的情形，我们证明了自相似坍缩解族连续依赖于哈密顿量，并且当哈密顿量接近一定值时，自相似解渐近地接近相对平衡.此外，我们还证明了四点涡系统相对平衡点的存在性。本文还研究了一个七点涡的非均匀涡强度的例子，并给出了数值结果。
英文摘要：
We study self-similar solutions of the point-vortex system. The explicit formula for self-similar solutions has been obtained for the three point-vortex problem and for a specific example of the four and five point-vortex problems. We see that the families consisting of these self-similar collapsing solutions are described by one-parameter families, and their collapse time and Hamiltonian are also expressed by functions of the same parameter. Then, the configurations at limit points of the parameter are in relative equilibria. For the many-vortex problem, we investigate the point-vortex system with the help of numerical computations. In particular, considering the case that $N - 1$ point vortices have a uniform vortex strength, we show that families of self-similar collapsing solutions continuously depend on the Hamiltonian and the self-similar solutions asymptotically approach relative equilibria as the Hamiltonian gets close to certain values. In addition, we prove the existence of relative equilibria for the four point-vortex system. We also investigate an example of seven point vortices with non-uniform vortex strengths and give numerical results for it.
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