The global Cauchy problem for the NLS with higher order anisotropic dispersion

We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M_{p,q}^s({\mathbb{R}}^d)$, where $q=1$ and $s\ge 0$ or $q\in (1,\mathrm{\infty }]$ and $s>\frac{d}{q^{\mathrm{\prime }}}$ for a nonlinear Schr"odinger equation with higher order anisotropic dispersion and algebraic nonlinearities.