Approximation theorem for the self-focusing Nonlinear Schr\"odinger   Equation and for the periodic curves in ${\bf R}^3$
  It is shown, that any sufficiently smooth periodic solution of the self-focusing Nonlinear Schr\"odinger equation can be approximated by periodic finite-gap ones with an arbitrary small error. As a corollary an analogous result for the motion of closed curves in ${\Bbb R}^3$ guided by the Filament equation is proved. This equation describes the dynamics of very thin filament vortices in a fluid. 
Piotr G. Grinevich