Spacetime quasiperiodic solutions to a nonlinear Schrodinger equation on Z

We consider a discrete non-linear Schrodinger equation on Z and show that, after adding a small potential localized in the time-frequency space, one can construct a three-parametric family of non-decaying spacetime quasiperiodic solutions to this equation. The proof is based on the Craig-Wayne-Bourgain method combined with recent techniques of dealing with Anderson localization for two-dimensional quasiperiodic operators with degenerate frequencies.