Vortex Filament Equation and Non-linear Schrdinger Equation

Abstract

In 1906, da Rios, a student of Leivi-Civita, wrote a master’s thesis modeling the motion of a vortex in a viscous fluid by the motion of a curve propagating in R3, in the direction of its binormal with a speed equal to its curvature. Much later, in 1971 Hasimoto showed the equivalence of this system with the non-linear Schr¨odinger equation (NLS qt = i(qss +12 |q|2q). In this paper, we use the same idea as Terng used in her lecture notes but different technique to extend the above relation to the case of S3, and obtained an analogous equation that qt = i[qss + (12 |q|2 + 1)q].