  We consider the binormal flow equation, which is a model for the dynamics of
vortex filaments in Euler equations. Geometrically it is a flow of curves in
three dimensions, explicitly connected to the 1-D Schr\"odinger map with values
on the 2-D sphere, and to the 1-D cubic Schr\"odinger equation. Although these
equations are completely integrable we show the existence of an unbounded
growth of the energy density. The density is given by the amplitude of the high
frequencies of the derivative of the tangent vectors of the curves, thus giving
information of the oscillation at small scales. In the setting of vortex
filaments the variation of the tangent vectors is related to the derivative of
the direction of the vorticity, that according to the
Constantin-Fefferman-Majda criterion plays a relevant role in the possible
development of singularities for Euler equations.
