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

Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$