Fractional Schrödinger dynamics and decoherence

Physica D: Nonlinear Phenomena - Tập 332 - Trang 41-54 - 2016
Kay Kirkpatrick1, Yanzhi Zhang2
1University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 W. Green Street Urbana, IL 61801, United States
2Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, United States

Tài liệu tham khảo

Gaididei, 1997, Effects of nonlocal dispersive interactions on self-trapping excitations, Phys. Rev. E, 55, 6141, 10.1103/PhysRevE.55.6141 Laskin, 2002, Fractional Schrödinger equation, Phys. Rev. E, 66, 10.1103/PhysRevE.66.056108 Zoia, 2007, Fractional Laplacian in bounded domains, Phys. Rev. E, 76, 10.1103/PhysRevE.76.021116 Michelitsch, 2013, The fractional Laplacian as a limiting case of a self-similar spring model and application to n-dimensional anomalous diffusion, Fract. Calc. Appl. Anal., 16, 827, 10.2478/s13540-013-0052-5 Mingaleev, 1999, Models for energy and charge transport and storage in biomolecules, J. Biol. Phys., 25, 41, 10.1023/A:1005152704984 Fröhlich, 2007, Boson stars as solitary waves, Comm. Math. Phys., 274, 1, 10.1007/s00220-007-0272-9 Lenzmann, 2007, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10, 43, 10.1007/s11040-007-9020-9 Kirkpatrick, 2013, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317, 563, 10.1007/s00220-012-1621-x Secchi, 2014, Soliton dynamics for fractional Schrödinger equations, Appl. Anal., 93, 1702, 10.1080/00036811.2013.844793 Hu, 2000, Schrödinger equations with fractional Laplacians, Appl. Math. Optim., 42, 281, 10.1007/s002450010014 Samko, 1993 Oldham, 2006 Pitaevskii, 2003 Bao, 2006, Dynamics of rotating Bose–Einstein condensates and its efficient and accurate numerical computation, SIAM J. Appl. Math., 66, 758, 10.1137/050629392 Zhang, 2007, Dynamics of the center of mass in rotating Bose–Einstein condensates, Appl. Numer. Math., 57, 697, 10.1016/j.apnum.2006.07.011 Frank, 2013, On the uniqueness and nondegeneracy of ground states of (−Δ)sQ+Q−Qα+1=0 in R, Acta Math., 210, 261, 10.1007/s11511-013-0095-9 Frank, 2015, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. Guo, 2013, Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg–Landau equation, Fract. Calc. Appl. Anal., 16, 226, 10.2478/s13540-013-0014-y Guo Cho, 2015, On the finite time blowup for mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145, 467, 10.1017/S030821051300142X Smith, 1991, 108 Fröhlich, 2004, Dynamics of solitary waves in an external potential, Comm. Math. Phys., 250, 613, 10.1007/s00220-004-1128-1 Benci, 2010, The nonlinear Schrödinger equation: Soliton dynamics, J. Differential Equations, 249, 3312, 10.1016/j.jde.2010.09.026 Bronski, 2000, Soliton dynamics in a potential, Math. Res. Lett., 7, 329, 10.4310/MRL.2000.v7.n3.a7 García-Ripoll, 2001, Construction of exact solutions by spatial translations in inhomogeneous nonlinear Schrödinger equations, Phys. Rev. E, 64, 10.1103/PhysRevE.64.056602 Zhang, 2010, Numerical study of vortex interactions in Bose–Einstein condensation, Commun. Comput. Phys., 8, 327, 10.4208/cicp.2009.09.104 Caliari, 2010, Numerical computation of soliton dynamics for NLS equations in a driving potential, Electron. J. Differential Equations, 89, 1 Ervin, 2007, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equations, SIAM J. Numer. Anal., 45, 572, 10.1137/050642757 Meerschaert, 2006, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 249, 10.1016/j.jcp.2005.05.017 Roop, 2006, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, J. Comput. Appl. Math., 193, 243, 10.1016/j.cam.2005.06.005 Cordoba, 2008, A note on the interface dynamics for convection in porous media, Physica D, 237, 1488, 10.1016/j.physd.2008.03.042 Berselli, 2014, Local solvability and turning for the inhomogeneous Muskat problem, Interfaces Free Bound., 16, 175, 10.4171/IFB/317 Granero-Belinchón, 2014, An aggregation equation with a nonlocal flux, Nonlinear Anal.-Theor., 108, 260, 10.1016/j.na.2014.05.018 Bueno-Orovio, 2014, Fourier spectral methods for fractional-in-space reaction–diffusion equations, BIT, 54, 937, 10.1007/s10543-014-0484-2 Amore, 2010, Collocation method for fractional quantum mechanics, J. Math. Phys., 51, 10.1063/1.3511330 Cai, 2001, Dispersive wave turbulence in one dimension, Physica D, 152–153, 551, 10.1016/S0167-2789(01)00193-2 Majda, 1997, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 6, 9, 10.1007/BF02679124 Zakharov, 2001, Wave turbulence in one-dimensional models, Physica D, 152–153, 573, 10.1016/S0167-2789(01)00194-4 Du, 2012, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 667, 10.1137/110833294 Duo, 2015, Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18, 321, 10.4208/cicp.300414.120215a Cordoba, 2007, Analytical behavior of 2D incompressible flow in porous media, J. Math. Phys., 48, 10.1063/1.2404593 Granero-Belinchón, 2015, On a nonlocal analog of the Kuramoto–Sivashinsky equation, Nonlinearity, 28, 1103, 10.1088/0951-7715/28/4/1103 Duo, 2016, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71, 2257, 10.1016/j.camwa.2015.12.042 Strang, 1968, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 506, 10.1137/0705041 Marchuk, 1990, Splitting and alternating direction methods, 10.1016/S1570-8659(05)80035-3 K. Kirkpatrick, Y. Zhang, Ground states of fractional Schrödinger equation, preprint. Bao, 2013, A simple and efficient numerical method for computing the dynamics of rotating Bose–Einstein condensates via a rotating Lagrangian coordinate, SIAM J. Sci. Comput., 35, A2671, 10.1137/130911111 Sulem, 1999 Zaslavsky, 2007, Dynamics of the chain of forced oscillators with long-range interaction: From synchronization to chaos, Chaos, 17, 10.1063/1.2819537 Korabel, 2007, Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction, Physica A, 378, 223, 10.1016/j.physa.2006.10.041 Tarasov, 2006, Continuous limit of discrete systems with long-range interaction, J. Phys. A: Math. Gen., 39, 14895, 10.1088/0305-4470/39/48/005