Approximation theorem for the self-focusing Nonlinear Schrِdinger Equation and for the periodic curves in R3

It is shown that any sufficiently smooth periodic solution of the self-focusing Nonlinear Schrِdinger Equation can be approximated by finite-gap ones with an arbitrary small error. As a corollary an analogous result for the motion of closed curves in R3 guided by the Filament equation is proved. This equation describes the dynamics of very thin filament vortices in a fluid.