Global well-posedness of the Gross--Pitaevskii and cubic-quintic nonlinear Schr\"odinger equations with non-vanishing boundary conditions

We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schr\"odinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.

[1]  Kh. I. Pushkarov,et al.  Solitary Clusters of Spin Deviations and Lattice Deformation in an Anharmonic Ferromagnetic Chain , 1984 .

[2]  STABILITY OF ENERGY-CRITICAL NONLINEAR SCHR¨ ODINGER EQUATIONS IN HIGH DIMENSIONS , 2005, math/0507005.

[3]  K. Nakanishi,et al.  Scattering theory for the Gross-Pitaevskii equation in three dimensions , 2008, 0803.3208.

[4]  K. Nakanishi,et al.  Scattering for the Gross-Pitaevskii equation , 2005 .

[5]  R. Killip,et al.  Global well-posedness and scattering for the defocusing quintic NLS in three dimensions , 2011, 1102.1192.

[6]  Jean-Claude Saut,et al.  Travelling Waves for the Gross-Pitaevskii Equation II , 2007, 0711.2408.

[7]  R. Killip,et al.  Nonlinear Schrodinger Equations at Critical Regularity , 2013 .

[8]  K. Yajima.  Existence of solutions for Schrödinger evolution equations , 1987 .

[9]  I. V. Barashenkov,et al.  Stability of the soliton-like “bubbles” , 1989 .

[10]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[11]  David Chiron,et al.  Traveling Waves for Nonlinear Schrödinger Equations with Nonzero Conditions at Infinity , 2012, Archive for Rational Mechanics and Analysis.

[12]  Robert S. Strichartz,et al.  Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations , 1977 .

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

[14]  Frisch,et al.  Transition to dissipation in a model of superflow. , 1992, Physical review letters.

[15]  Superfluidity of helium II near the λ point , 1976 .

[16]  Kh. I. Pushkaeov,et al.  Solitary excitations in molecular chains in the presence of vibronic states , 1986 .

[17]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

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

[19]  M. Visan.  Global Well-posedness and Scattering for the Defocusing Cubic nonlinear Schrödinger equation in Four Dimensions , 2012 .

[20]  B. Wang.  The Cauchy problem for the nonlinear Schrödinger equations involving derivative terms in one spatial dimension , 2003 .

[21]  Patrick Gérard,et al.  The Cauchy problem for the Gross-Pitaevskii equation , 2006 .

[22]  A. B. Datseff.  On the nonlinear schrödinger equation , 2009 .

[23]  J. Ginibre,et al.  Smoothing properties and retarded estimates for some dispersive evolution equations , 1992 .

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

[25]  T. Tao,et al.  Endpoint Strichartz estimates , 1998 .

[26]  C. Gallo.  Schrödinger group on Zhidkov spaces , 2004, Advances in Differential Equations.

[27]  Terence Tao,et al.  The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities , 2005 .

[28]  N. Bogolyubov.  On the theory of superfluidity , 1947 .

[29]  I. Ventura.  Theory of Superfluidity , 1979 .