  We consider the problem of transfer of energy to high frequencies in a
quasilinear Schr\"odinger equation with sublinear dispersion, on the one
dimensional torus. We exhibit initial data undergoing finite but arbitrary
large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev
spaces of high regularity, but at a later time becomes arbitrary large. We
develop a novel mechanism producing instability, which is based on extracting,
via paradifferential normal forms, an effective equation driving the dynamics
whose leading term is a non-trivial transport operator with non-constant
coefficients. We prove that such operator is responsible for energy cascades
via a positive commutator estimate inspired by Mourre's commutator theory.
