  We study the double cascade of energy and wave action in a local model of
superfluid vortex filaments. The model is obtained from a truncated expansion
of the 2D Local Induction Approximation and it is shown to support six-wave
interactions. We argue that, because of the uncertainty in the vortex core
profile, this model has the same status of validity as the traditionally used
Biot-Savart model with cutoff, but it has advantage of being much simpler. Our
minimal model leads to a wave kinetic equation for which we predict existence
of two distinct power-law scaling in the spectrum, corresponding to a direct
cascade of energy and an inverse one of wave action. Direct numerical
simulations confirm the theoretical predictions in the weak turbulence regime.
