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 equations i∂tu=−ε u+ 2u+ P ∂α x u |α| 2, ∂α x ¯u |α| 2 , t∈ R, x ∈ Rn, where ε ∈ {−1, 0, 1}, n 2 denotes the spatial dimension and P(·) is a polynomial excluding constant and linear terms. © 2006 Elsevier Inc. All rights reserved.