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Preface
Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential
1. | IMT, UMR CNRS 5219, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France |
2. | Department of Mathematics, National University of Singapore, Singapore 119076, Singapore |
3. | IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France, France |
References:
[1] |
G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), 115-124.
doi: 10.1093/imanum/13.1.115. |
[2] |
W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comp. Phys., 187 (2003), 318-342.
doi: 10.1016/S0021-9991(03)00102-5. |
[3] |
W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), 767-782.
doi: 10.1142/S0218202505000534. |
[4] |
W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 2010-2028.
doi: 10.1137/030601211. |
[5] |
W. Bao and J. Shen, A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates, J. Comput. Phys., 227 (2008), 9778-9793.
doi: 10.1016/j.jcp.2008.07.017. |
[6] |
N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differential Equations, 245 (2008), 154-200. |
[7] |
N. Ben Abdallah, F. Méhats and O. Pinaud, Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement,, SIAM J. Math. Anal., 36 (): 986.
doi: 10.1137/S0036141003437915. |
[8] |
N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), 189-199.
doi: 10.1137/040614554. |
[9] |
B. Bidéaray-Fesquet, F. Castella and P. Degond, From Bloch model to the rate equations, Discrete Contin. Dyn. Syst., 11 (2004), 1-26.
doi: 10.3934/dcds.2004.11.1. |
[10] |
B. Bidéaray-Fesquet, F. Castella, E. Dumas and M. Gisclon, From Bloch model to the rate equations. II. The case of almost degenerate energy levels, Math. Models Methods Appl. Sci., 14 (2004), 1785-1817.
doi: 10.1142/S0218202504003829. |
[11] |
J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, 122 (1994), 77-118. |
[12] |
F. Bornemann, "Homogenization in Time of Singularly Perturbed Mechanical Systems," Lecture Notes in Mathematics, 1687, Springer-Verlag, Berlin, 1998. |
[13] |
B. M. Caradoc-Davis, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), 895-898.
doi: 10.1103/PhysRevLett.83.895. |
[14] |
F. Castella, P. Degond and T. Goudon, Diffusion dynamics of classical systems driven by an oscillatory force, J. Stat. Phys., 124 (2006), 913-950.
doi: 10.1007/s10955-006-9071-5. |
[15] |
F. Castella, P. Degond and T. Goudon, Large time dynamics of a classical system subject to a fast varying force, Comm. Math. Phys., 276 (2007), 23-49.
doi: 10.1007/s00220-007-0339-7. |
[16] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lect. Notes Math., 10, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, R.I., 2003. |
[17] |
F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field, Comm. Math. Phys., 292 (2009), 829-870. |
[18] |
G. F. Dell'Antonio and L. Tenuta, Semiclassical analysis of constrained quantum systems, J. Phys. A, 37 (2004), 5605-5624.
doi: 10.1088/0305-4470/37/21/007. |
[19] |
C. M. Dion and E. Cances, Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), 046706.
doi: 10.1103/PhysRevE.67.046706. |
[20] |
D. Funaro, "Polynomial Approximations of Differential Equations," Lecture Notes in Physics, New Series m: Monographs, 8, Springer-Verlag, Berlin, 1992. |
[21] |
E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.
doi: 10.1016/S0021-7824(97)89959-X. |
[22] |
R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. Chronicle, 15 (1973), 423. |
[23] |
B. Helffer, "Théorie Spectrale pour des Opérateurs Globalement Elliptiques," Astérisque, 112, Société Mathématique de France, Paris, 1984. |
[24] |
B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians," Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. |
[25] |
D. Lannes, Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum, Adv. Differential Equations, 6 (2001), 731-768. |
[26] |
G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differential Equations, 187 (2003), 106-183.
doi: 10.1016/S0022-0396(02)00037-2. |
[27] |
L. Pitaevskii and S. Stringari, "Bose-Einstein Condensation," International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. |
[28] |
M. P. Robinson, G. Fairweather and B. M. Herbst, On the numerical solution of the cubic Schrödinger equation in one space variable, J. Comput. Phys., 104 (1993), 277-284.
doi: 10.1006/jcph.1993.1029. |
[29] |
J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Appl. Math. Sci., 59, Springer-Verlag, New York, 1985. |
[30] |
J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Appl. Math. Sci., 59, Springer, New York, 2007. |
[31] |
S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.
doi: 10.1006/jdeq.1994.1157. |
[32] |
G. Szegö, "Orthogonal Polynomials," 4th edition, Amer. Math. Soc., Colloq. Publ., Vol. XXIII, AMS, Providence, R.I., 1975. |
[33] |
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55 (1984), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[34] |
J. Wachsmuth and S. Teufel, Constrained quantum systems as an adiabatic problem, Phys. Rev. A, 82 (2010), 022112.
doi: 10.1103/PhysRevA.82.022112. |
show all references
References:
[1] |
G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), 115-124.
doi: 10.1093/imanum/13.1.115. |
[2] |
W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comp. Phys., 187 (2003), 318-342.
doi: 10.1016/S0021-9991(03)00102-5. |
[3] |
W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments, Math. Models Meth. Appl. Sci., 15 (2005), 767-782.
doi: 10.1142/S0218202505000534. |
[4] |
W. Bao and J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 2010-2028.
doi: 10.1137/030601211. |
[5] |
W. Bao and J. Shen, A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates, J. Comput. Phys., 227 (2008), 9778-9793.
doi: 10.1016/j.jcp.2008.07.017. |
[6] |
N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differential Equations, 245 (2008), 154-200. |
[7] |
N. Ben Abdallah, F. Méhats and O. Pinaud, Adiabatic approximation of the Schrödinger-Poisson system with a partial confinement,, SIAM J. Math. Anal., 36 (): 986.
doi: 10.1137/S0036141003437915. |
[8] |
N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential, SIAM J. Math. Anal., 37 (2005), 189-199.
doi: 10.1137/040614554. |
[9] |
B. Bidéaray-Fesquet, F. Castella and P. Degond, From Bloch model to the rate equations, Discrete Contin. Dyn. Syst., 11 (2004), 1-26.
doi: 10.3934/dcds.2004.11.1. |
[10] |
B. Bidéaray-Fesquet, F. Castella, E. Dumas and M. Gisclon, From Bloch model to the rate equations. II. The case of almost degenerate energy levels, Math. Models Methods Appl. Sci., 14 (2004), 1785-1817.
doi: 10.1142/S0218202504003829. |
[11] |
J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, 122 (1994), 77-118. |
[12] |
F. Bornemann, "Homogenization in Time of Singularly Perturbed Mechanical Systems," Lecture Notes in Mathematics, 1687, Springer-Verlag, Berlin, 1998. |
[13] |
B. M. Caradoc-Davis, R. J. Ballagh and K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), 895-898.
doi: 10.1103/PhysRevLett.83.895. |
[14] |
F. Castella, P. Degond and T. Goudon, Diffusion dynamics of classical systems driven by an oscillatory force, J. Stat. Phys., 124 (2006), 913-950.
doi: 10.1007/s10955-006-9071-5. |
[15] |
F. Castella, P. Degond and T. Goudon, Large time dynamics of a classical system subject to a fast varying force, Comm. Math. Phys., 276 (2007), 23-49.
doi: 10.1007/s00220-007-0339-7. |
[16] |
T. Cazenave, "Semilinear Schrödinger Equations," Courant Lect. Notes Math., 10, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, R.I., 2003. |
[17] |
F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field, Comm. Math. Phys., 292 (2009), 829-870. |
[18] |
G. F. Dell'Antonio and L. Tenuta, Semiclassical analysis of constrained quantum systems, J. Phys. A, 37 (2004), 5605-5624.
doi: 10.1088/0305-4470/37/21/007. |
[19] |
C. M. Dion and E. Cances, Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. Rev. E, 67 (2003), 046706.
doi: 10.1103/PhysRevE.67.046706. |
[20] |
D. Funaro, "Polynomial Approximations of Differential Equations," Lecture Notes in Physics, New Series m: Monographs, 8, Springer-Verlag, Berlin, 1992. |
[21] |
E. Grenier, Oscillatory perturbations of the Navier-Stokes equations, J. Math. Pures Appl. (9), 76 (1997), 477-498.
doi: 10.1016/S0021-7824(97)89959-X. |
[22] |
R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. Chronicle, 15 (1973), 423. |
[23] |
B. Helffer, "Théorie Spectrale pour des Opérateurs Globalement Elliptiques," Astérisque, 112, Société Mathématique de France, Paris, 1984. |
[24] |
B. Helffer and F. Nier, "Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians," Lecture Notes in Mathematics, 1862, Springer-Verlag, Berlin, 2005. |
[25] |
D. Lannes, Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum, Adv. Differential Equations, 6 (2001), 731-768. |
[26] |
G. Métivier and S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations, J. Differential Equations, 187 (2003), 106-183.
doi: 10.1016/S0022-0396(02)00037-2. |
[27] |
L. Pitaevskii and S. Stringari, "Bose-Einstein Condensation," International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. |
[28] |
M. P. Robinson, G. Fairweather and B. M. Herbst, On the numerical solution of the cubic Schrödinger equation in one space variable, J. Comput. Phys., 104 (1993), 277-284.
doi: 10.1006/jcph.1993.1029. |
[29] |
J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Appl. Math. Sci., 59, Springer-Verlag, New York, 1985. |
[30] |
J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Appl. Math. Sci., 59, Springer, New York, 2007. |
[31] |
S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512.
doi: 10.1006/jdeq.1994.1157. |
[32] |
G. Szegö, "Orthogonal Polynomials," 4th edition, Amer. Math. Soc., Colloq. Publ., Vol. XXIII, AMS, Providence, R.I., 1975. |
[33] |
T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55 (1984), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[34] |
J. Wachsmuth and S. Teufel, Constrained quantum systems as an adiabatic problem, Phys. Rev. A, 82 (2010), 022112.
doi: 10.1103/PhysRevA.82.022112. |
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