论文标题
双曲机中常规多边形的涡旋丝方方程
Vortex Filament Equation for a regular polygon in the hyperbolic plane
论文作者
论文摘要
本文的目的是双重的。首先，我们显示了双曲线空间中常规平面多边形的涡旋丝方程（VFE）的演变。与欧几里得空间不同，平面多边形是开放的，并且其两端呈指数增长，从数值的角度来看，问题更具挑战性。但是，在固定的边界条件下，有限的差异方案和四阶runge-kutta方法及时，我们表明数值解决方案与从代数技术获得的解决方案完全一致。其次，与欧几里得情况一样，我们声称，在无穷小时，对于平面多边形的VFE的演变可以描述为几个单角初始数据的叠加。结果，我们不仅可以计算平面多边形的质量中心的速度，而且这种关系还允许我们将其任何角的时间演变与欧几里得情况下的时间演变进行比较。
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
