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

Sharp global well-posedness for non-elliptic derivative Schr\"odinger equations with small rough data