  We consider Nonlinear Schrodinger type equations on $S^1$. In this paper, we
obtain polynomial bounds on the growth in time of high Sobolev norms of their
solutions. The key is to derive an iteration bound based on a frequency
decomposition of the solution. This iteration bound is different than the one
used earlier in the work of Bourgain, and is less dependent on the structure of
the nonlinearity. We first look at the defocusing NLS equation with
nonlinearity of degree $\geq 5$. For the quintic NLS, Bourgain derives stronger
bounds using different techniques. However, our approach works for higher
nonlinearities, where the techniques of Bourgain don't seem to apply.
Furthermore, we study variants of the defocusing cubic NLS in which the
complete integrability is broken. Among this class of equations, we consider in
particular the Hartree Equation, with sufficiently regular convolution
potential. For most of the equations that come from modifying the defocusing
cubic NLS, we obtain better bounds than for the other equations due to the fact
that we can use higher modified energies as in the work of the I-Team.
