  We apply the proper orthogonal decomposition (POD) to the nonlinear
Schr\"odinger (NLS) equation to derive a reduced order model. The NLS equation
is discretized in space by finite differences and is solved in time by
structure preserving symplectic mid-point rule. A priori error estimates are
derived for the POD reduced dynamical system. Numerical results for one and two
dimensional NLS equations, coupled NLS equation with soliton solutions show
that the low-dimensional approximations obtained by POD reproduce very well the
characteristic dynamics of the system, such as preservation of energy and the
solutions.
