Periodic fourth-order cubic NLS: Local well-posedness and Non-squeezing property

In this paper, we consider the cubic fourth-order nonlinear Schr\"odinger equation (4NLS) under the periodic boundary condition. We prove two results. One is the local well-posedness in $H^s$ with $-1/3 \le s < 0$ for the Cauchy problem of the Wick ordered 4NLS. The other one is the non-squeezing property for the flow map of 4NLS in the symplectic phase space $L^2(\mathbb{T})$. To prove the former we used the ideas introduced in [Takaoka and Tsutsumi 2004] and [Nakanish et al 2010], and to prove the latter we used the ideas in [Colliander et al 2005].