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 equationi∂tu+Δ2u+|u|2u=0i∂tu+Δ2u+|u|2u=0 in arbitrary space dimension RnRn for arbitrary initial data. We prove that the equation is globally well-posed when n⩽8n⩽8 and ill-posed when n⩾9n⩾9, with the additional important information that scattering holds true when 5⩽n⩽85⩽n⩽8.