Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equationsi∂tu=−εΔu+Δ2u+P((∂xαu)|α|⩽2,(∂xαu¯)|α|⩽2),t∈R,x∈Rn, where ε∈{−1,0,1}ε∈{−1,0,1}, n⩾2n⩾2 denotes the spatial dimension and P(⋅)P(⋅) is a polynomial excluding constant and linear terms.