论文标题
$ n $ point涡流系统中的自相似动作和相关的相对平衡
Self-similar motions and related relative equilibria in the $N$-point vortex system
论文作者
论文摘要
我们研究点涡流系统的自相似解决方案。对于三个点涡流问题，已经获得了自相似解决方案的明确公式，以及四个和五个涡流问题的特定示例。我们看到，由一个参数家族描述了由这些自相似崩溃的解决方案组成的家庭，它们的崩溃时间和汉密尔顿人也通过相同参数的功能表达。然后，参数的限制点的配置为相对平衡。对于多涡流问题，我们在数值计算的帮助下研究了点涡流系统。特别是，考虑到$ n-1 $ point涡流具有均匀的涡旋强度的情况，我们表明，自相似折叠式解决方案的家庭不断取决于哈密顿量和自相似的解决方案相对均等的均等均衡，因为哈密顿量接近一定值。此外，我们证明了四点涡流系统的相对平衡存在。我们还研究了具有不均匀涡旋强度的七个涡旋的示例，并为其给出数值结果。
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.
