Long-time Anderson Localization for the Nonlinear quasi-periodic Schrödinger Equation on ${\mathbb{Z}}^d$

Using a Birkhoff normal form transform to impede mode transfer in a finite “barrier”, we prove localization of arbitrary ${\mathrm{\ell }}^2$ data for polynomially long time for the nonlinear quasi-periodic Schrödinger equation on ${\mathbb{Z}}^d$.