The cubic fourth-order Schrödinger equation

Fourth-order Schrodinger 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 Schrodinger equation i∂tu+Δ2u+|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.

[1]  Remi Carles.  Geometric Optics and Instability for Semi-Classical Schrödinger Equations , 2007 .

[2]  T. Tao,et al.  Ill-posedness for nonlinear Schrodinger and wave equations , 2003, math/0311048.

[3]  V. Karpman,et al.  Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion , 2000 .

[4]  W. Strauss,et al.  Decay and scattering of solutions of a nonlinear Schrödinger equation , 1978 .

[5]  R. Killip,et al.  The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher , 2008, 0804.1018.

[6]  Y. Meyer,et al.  Inégalités de Sobolev précisées , 1997 .

[7]  Laurent Thomann.  Instabilities for supercritical Schr\"odinger equations in analytic manifolds , 2007, 0707.1785.

[8]  Terence Tao,et al.  The cubic nonlinear Schr\"odinger equation in two dimensions with radial data , 2007, 0707.3188.

[9]  J. Segata.  Remark on well-posedness for the fourth order nonlinear Schrödinger type equation , 2004 .

[10]  Zhaohui Huo,et al.  The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament , 2005 .

[11]  Karpman,et al.  Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  Luis Vega,et al.  Oscillatory integrals and regularity of dispersive equations , 1991 .

[13]  Tosio Kato.  On nonlinear Schrödinger equations, II.HS-solutions and unconditional well-posedness , 1995 .

[14]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

[15]  Terence Tao,et al.  Minimal-mass blowup solutions of the mass-critical NLS , 2006, math/0609690.

[16]  F. Merle,et al.  Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation , 2006, math/0610801.

[17]  L. Hsiao,et al.  Wellposedness for the fourth order nonlinear Schrödinger equations , 2006 .

[18]  J. Segata.  Modified wave operators for the fourth‐order non‐linear Schrödinger‐type equation with cubic non‐linearity , 2006 .

[19]  B. Pausader.  The focusing energy-critical fourth-order Schrödinger equationwith radial data , 2009 .

[20]  W. Strauss,et al.  Analyticity of the nonlinear scattering operator , 2009 .

[21]  M. Visan,et al.  Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4 , 2007 .

[22]  Gadi Fibich,et al.  Self-Focusing with Fourth-Order Dispersion , 2002, SIAM J. Appl. Math..

[23]  Laurent Thomann.  Geometric and projective instability for the Gross-Pitaevski equation , 2007, Asymptot. Anal..

[24]  瀬片 純市.  Well-posedness for the Fourth Order Nonlinear Schrodinger Type Equation Related to the Vortex Filament (流体と気体の数学解析 研究集会報告集) , 2003 .

[25]  Carlos E. Kenig,et al.  Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case , 2006 .

[26]  Jean-Claude Saut,et al.  Dispersion estimates for fourth order Schr?odinger equations , 2000 .

[27]  Benoit Pausader,et al.  Global Well-Posedness for Energy Critical Fourth-Order Schrodinger Equations in the Radial Case , 2007 .

[28]  S. Keraani.  On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations , 2001 .

[29]  M. Visan.  The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions , 2005, math/0508298.

[30]  G. Lebeau.  Non linear optic and supercritical wave equation , 1980 .

[31]  G. Lebeau.  Perte de régularité pour les équations d’ondes sur-critiques , 2005 .

[32]  P. Gérard,et al.  High frequency approximation of solutions to critical nonlinear wave equations , 1999 .

[33]  L. Hsiao,et al.  Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces , 2007 .

[34]  T. Tao.  Nonlinear dispersive equations : local and global analysis , 2006 .

[35]  Global well-posedness and scattering for the energy-critical nonlinear Schr\"odinger equation in R^3 , 2004, math/0402129.

[36]  Gadi Fibich,et al.  Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion , 2003 .