Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4592470 | Journal of Functional Analysis | 2009 | 45 Pages |
Abstract
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Benoit Pausader,