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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.

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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).

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  • $\begingroup$ 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? $\endgroup$ – Kei May 15 '19 at 21:00
  • $\begingroup$ 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. $\endgroup$ – Carlo Beenakker May 15 '19 at 21:02

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