Wellposedness for the fourth order nonlinear Schrödinger equations

We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1D i∂tu = ∂4 xu +P ∂α x u |α| 2, ∂α x ¯u |α| 2 , t,x∈ R, where P(·) is a polynomial excluding constant or linear terms. © 2005 Elsevier Inc. All rights reserved