The cubic fourth-order Schrödinger equation

Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth-order Schrödinger equation i∂tu + 2u + |u|2u = 0 in arbitrary space dimension Rn for arbitrary initial data. We prove that the equation is globally well-posed when n 8 and ill-posed when n 9, with the additional important information that scattering holds true when 5 n 8. © 2008 Elsevier Inc. All rights reserved