论文标题
在Frenet框架中表达的一般流的Hasimoto转换
Hasimoto Transformation of General Flows Expressed in the Frenet Frame
论文作者
论文摘要
$ \ mathbb {r}^{3} $中的一维空间曲线是建模涡旋丝和生物软性物质的有用的非线性介质，能够支持各种波动。 Hasimoto转换定义了空间曲线的运动学演化与非线性标量方程在发展其内在曲线几何形状之间的映射。该映射非常健壮，能够转换在Frenet框架中表达的通用矢量场，从而产生了完全非线性的无线差异进化方程，其系数结构是由Frenet框架中流的坐标定义的。在本文中，我们将Hasimoto Map推广到空间曲线上定义的任意流，我们对几种现有的运动学流进行测试。此后，我们考虑长度和弯曲能量的时间动力学，以确保双重流动通常是长度的，并且弯曲能是脆弱的，在一般情况下不可能保守。
A one-dimensional space curve in $\mathbb{R}^{3}$ is a useful nonlinear medium for modeling vortex filaments and biological soft-matter capable of supporting a variety of wave motions. The Hasimoto transformation defines a mapping between the kinematic evolution of a space curve and nonlinear scalar equations evolving its intrinsic curve geometry. This mapping is quite robust and able to transform general vector fields expressed in the Frenet frame, resulting in a fully nonlinear integro-differential evolution equation, whose coefficient structure is defined by the coordinates of the flow in the Frenet frame. In this paper, we generalize the Hasimoto map to arbitrary flows defined on space curves, which we test against several existing kinematic flows. After this, we consider the time dynamics of length and bending energy to see that binormal flows are generally length preserving, and bending energy is fragile and unlikely to be conserved in the general case.
