  In this paper we consider two classes of resonant Hamiltonian PDEs on the
circle with non-convex (respect to actions) first order resonant Hamiltonian.
We show that, for appropriate choices of the nonlinearities we can find
time-independent linear potentials that enable the construction of solutions
that undergo a prescribed growth in the Sobolev norms. The solutions that we
provide follow closely the orbits of a nonlinear resonant model, which is a
good approximation of the full equation. The non-convexity of the resonant
Hamiltonian allows the existence of fast diffusion channels along which the
orbits of the resonant model experience a large drift in the actions in the
optimal time. This phenomenon induces a transfer of energy among the Fourier
modes of the solutions which in turn is responsible for the growth of higher
order Sobolev norms.
