The cubic fourth-order Schr_dinger equation

摘要

Fourth-order Schr_dinger 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 Schr_dinger equation iδ_tu + Δ~2u + |u|~2u =0 in arbitrary space dimension R~n 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.