# Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $$\chi(t,s)$$ is a curve in $$\mathbb R^3$$, related to the cubic Schrödinger equation?

Note 1. I've asked a more general question at Survey on the vortex filament equation.

If $$\kappa$$ and $$\tau$$ are, respectively, the curvature and torsion of $$\gamma$$, and one defines the wave function $$\psi=\kappa\exp\left(i\int \tau\,dx\right)$$, then $$\psi$$ satisfies the nonlinear (cubic) Schrödinger equation [1,2,3] $$-i\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+\tfrac{1}{2}|u|^2 u.$$

[1] H. Hasimoto, A soliton on a vortex filament (1972).
[2] N. Koiso, The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space (1997).
[3] D.D. Holm and S.N. Stechmann, Hasimoto transformation and vortex soliton motion driven by fluid helicity (2004).

• Thank you. I've checked those references. They are really helpful. I've just an additional question: what is the physical meaning of that wave function $\psi$? That is, physically, why does the soliton ansatz of considering $\psi$ connect the vortex filament equation and the Schroedinger one? – Kei May 15 '19 at 21:00
• it is a mathematical mapping of two unrelated physical problems, one problem from classical physics, the other from quantum physics; there is only a mathematical relation, no physics relation. – Carlo Beenakker May 15 '19 at 21:02