The Cubic Fourth-order Schrödinger Equation
نویسنده
چکیده
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 +∆ 2 u + |u|u = 0 in arbitrary space dimension R 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.
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تاریخ انتشار 2008