  We study the nonlinear Schr\"odinger equation (NLS) with bounded initial data
which does not vanish at infinity. Examples include periodic, quasi-periodic
and random initial data. On the lattice we prove that solutions are
polynomially bounded in time for any bounded data. In the continuum, local
existence is proved for real analytic data by a Newton iteration scheme. Global
existence for NLS with a regularized nonlinearity follows by analyzing a local
energy norm.
