Van Gorder Robert A
Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):057301. doi: 10.1103/PhysRevE.86.057301. Epub 2012 Nov 2.
We review two formulations of the fully nonlinear local induction equation approximating the self-induced motion of the vortex filament (in the local induction approximation), corresponding to the Cartesian and arc-length coordinate systems. The arc-length representation put forth by Umeki [Theor. Comput. Fluid Dyn. 24, 383 (2010)] results in a type of 1+1 derivative nonlinear Schrödinger (NLS) equation describing the motion of such a vortex filament. We obtain exact stationary solutions to this derivative NLS equation; such exact solutions are a rarity. These solutions are periodic in space and we determine the nonlinear dependence of the period on the amplitude.
我们回顾了两种完全非线性局部感应方程的形式，它们在局部感应近似下逼近涡旋丝的自感应运动，分别对应笛卡尔坐标系和弧长坐标系。梅木提出的弧长表示法[《理论与计算流体动力学》24, 383 (2010)]产生了一种1 + 1导数非线性薛定谔（NLS）方程，用于描述此类涡旋丝的运动。我们得到了这个导数NLS方程的精确驻波解；此类精确解十分罕见。这些解在空间上是周期性的，并且我们确定了周期对振幅的非线性依赖关系。
