Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schr\"odinger Equations in Multi-dimensional Spaces

We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2 u+P((\partial_x^\alpha u){\abs{\alpha}\ls 2}, (\partial_x^\alpha \bar{u}){\abs{\alpha}\ls 2}),\quad t\in \Real, x\in\Real^n, where $\text{eps}\in \{-1,0,1\}$, $n\text{gs}2$ denotes the spatial dimension and $P(\cdot )$ is a polynomial excluding constant and linear terms.