Effective Mass Theorems for Nonlinear Schrödinger Equations

We consider time-dependent nonlinear Schrödinger equations subject to smooth, lattice-periodic potentials plus additional confining potentials, slowly varying on the lattice scale. After an appropriate scaling we study the homogenization limit for vanishing lattice spacing. Assuming well prepared initial data, the resulting effective dynamics is governed by a homogenized nonlinear Schrödinger equation with an effective mass tensor depending on the initially chosen Bloch eigenvalue. The given results rigorously justify the use of the effective mass formalism for the description of Bose-Einstein condensates on optical lattices.