  We start a study of various nonlinear PDEs under the effect of a modulation
in time of the dispersive term. In particular in this paper we consider the
modulated non-linear Schr\"odinger equation (NLS) in dimension 1 and 2 and the
derivative NLS in dimension 1. We introduce a deterministic notion of
"irregularity" for the modulation and obtain local and global results similar
to those valid without modulation. In some situations, we show how the
irregularity of the modulation improves the well--posedness theory of the
equations. We develop two different approaches to the analysis of the effects
of the modulation. A first approach is based on novel estimates for the
regularising effect of the modulated dispersion on the non-linear term using
the theory of controlled paths. A second approach is an extension of a
Strichartz estimated first obtained by Debussche and Tsutsumi in the case of
the Brownian modulation for the quintic NLS.
