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?




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**Note 1.** I've asked a more general question at [Survey on the vortex filament equation](https://mathoverflow.net/questions/331539/survey-on-the-vortex-filament-equation).



Relationship between the vortex filament equation and the cubic Schrödinger 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](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/soliton-on-a-vortex-filament/1A2FF58DD1B8DE20509866A4713531F3) (1972).  

[2] N. Koiso, [The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space](https://projecteuclid.org/euclid.ojm/1200787341) (1997).  

[3] D.D. Holm and S.N. Stechmann, [Hasimoto transformation and
vortex soliton motion driven by fluid helicity](https://pdfs.semanticscholar.org/27b6/70827041e81a3f7b407386589f4a287602d5.pdf) (2004).


