  We show the sharp global well posedness for the Cauchy problem for the cubic
(quartic) non-elliptic derivative Schr\"odinger equations with small rough data
in modulation spaces $M^s_{2,1}(\mathbb{R}^n)$ for $n\ge 3$ ($n= 2$). In 2D
cubic case, using the Gabor frame, we get some time-global dispersive estimates
for the Schr\"odinger semi-group in anisotropic Lebesgue spaces, which include
a time-global maximal function estimate in the space
$L^2_{x_1}L^\infty_{x_2,t}$. By resorting to the smooth effect estimate
together with the dispersive estimates in anisotropic Lebesgue spaces, we show
that the cubic hyperbolic derivative NLS in 2D has a unique global solution if
the initial data in Feichtinger-Segal algebra or in weighted Sobolev spaces are
sufficiently small.
