# American Institute of Mathematical Sciences

December  2011, 4(4): 831-856. doi: 10.3934/krm.2011.4.831

## 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

Received  May 2011 Revised  September 2011 Published  November 2011

We consider the three dimensional Gross-Pitaevskii equation\break (GPE) describing a Bose-Einstein Condensate (BEC) which is highly confined in vertical $z$ direction. The confining potential induces high oscillations in time. If the confinement in the $z$ direction is a harmonic trap -- an approximation which is widely used in physical experiments -- the very special structure of the spectrum of the confinement operator implies that the oscillations are periodic in time. Based on this observation, it can be proved that the GPE can be averaged out with an error of order of $\epsilon$, which is the typical period of the oscillations. In this article, we construct a more accurate averaged model, which approximates the GPE up to errors of order $\mathcal{O}(\epsilon^2)$. Then, expansions of this model over the eigenfunctions (modes) of the confining operator $H_z$ in the $z$-direction are given in view of numerical applications. Efficient numerical methods are constructed to solve the GPE with cylindrical symmetry in 3D and the approximation model with radial symmetry in 2D, and numerical results are presented for various kinds of initial data.
Citation: Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic and Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831
##### 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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##### 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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