We focus on the existence of quasi-periodic, small amplitude, solutions for quasi-linear and fully nonlinear NLS equation on the torus in presence of a quasi-periodic forcing. We obtain the existence for a Cantor sets of frequencies which has asymptotically full measure as the perturbation parameter goes to zero.
We give a sketch of the proof which is based on a Nash-Moser iterative scheme on a scale of Banach spaces. In order to invert the linearized operator which has non constant coefficients up to the highest order derivatives, we perform a regularization procedure that conjugates it to a constant coefficients differential operator plus a bounded remainder. Hence we apply a KAM-like scheme to completely diagonalize it on a suitable Cantor set.