In this paper we study the existence of global small solutions of the Cauchy problem for the non-isotropically perturbed nonlinear Schr$ddot{o}$dinger equation: $iu_t +Delta u + |u|^alpha u+asum_i^du_{x_ix_ix_ix_i}=0$, where $a$ is real constant, $1le d < n$ is a integer, $alpha$ is a positive constant, and $x = (x_1,x_2,cdots , x_n)in R^n$. For some admissible $alpha$ we show the existence of global(almost global) solutions and we calculate the regularity of those solutions.

Keywords: non-isotropic Schr$ddot{o}$dinger equations, global solutions, almost global solutions, regularity