# American Institute of Mathematical Sciences

November  2015, 20(9): 2793-2817. doi: 10.3934/dcdsb.2015.20.2793

## Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time

 1 Centre de Mathématiques Appliquées, CNRS et Ecole Polytechnique, 91128 Palaiseau cedex, France 2 Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan 3 Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau cedex, France

Received  September 2014 Revised  March 2015 Published  November 2015

The aim of this paper is to perform a theoretical and numerical study on the dynamics of vortices in Bose-Einstein condensation in the case where the trapping potential varies randomly in time. We take a deterministic vortex solution as an initial condition for the stochastically fluctuated Gross-Pitaevskii equation, and we observe the influence of the stochastic perturbation on the evolution. We theoretically prove that up to times of the order of $\epsilon^{-2}$, the solution having the same symmetry properties as the vortex decomposes into the sum of a randomly modulated vortex solution and a small remainder, and we derive the equations for the modulation parameter. In addition, we show that the first order of the remainder, as $\epsilon$ goes to zero, converges to a Gaussian process. Finally, some numerical simulations on the dynamics of the vortex solution in the presence of noise are presented.
Citation: Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793
##### References:
 [1] F. Kh. Abdullaev, B. B. Baizakov and V. V. Konotop, Dynamics of a Bose-Einstein condensate in optical trap, in Nonlinearity and Disorder: Theory and Applications edited by F.Kh. Abdullaev, O. Bang and M.P. Soerensen, NATO Science Series, Kluwer Dodrecht, 45 (2001), 69-78. doi: 10.1007/978-94-010-0542-5_7. [2] F. Kh. Abdullaev, J. C. Bronski and G. Papanicolaou, Soliton perturbations and the random Kepler problem, Physica D, 135 (2000), 369-386. doi: 10.1016/S0167-2789(99)00118-9. [3] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956. [4] W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Models Methods Appl. Sci., 15 (2005), 1863-1896. doi: 10.1142/S021820250500100X. [5] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers, 1983. [6] C. C. Bradley, et al., Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75 (1995), 1687-1691. doi: 10.1103/PhysRevLett.75.1687. [7] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Providence, RI: American Mathematical Society/Courant Institute of Mathematical Sciences, 2003. [8] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM. J. Math. Anal., 39 (2007/08), 1070-1111. doi: 10.1137/050648389. [9] K. B. Davis, et al., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3974. doi: 10.1109/EQEC.1996.561567. [10] A. de Bouard and A. Debussche, Random modulation of solitons for the stochastic Korteweg-de Vries equation, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 251-278. doi: 10.1016/j.anihpc.2006.03.009. [11] A. de Bouard and A. Debussche, Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise, Electron. J. Probab., 14 (2009), 1727-1744. doi: 10.1214/EJP.v14-683. [12] A. de Bouard and R. Fukuizumi, Stochastic fluctuations in the Gross-Pitaevskii equation, Nonlinearity, 20 (2007), 2823-2844. doi: 10.1088/0951-7715/20/12/005. [13] A. de Bouard and R. Fukuizumi, Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation, Asymptot. Anal., 63 (2009), 189-235. [14] A. de Bouard and R. Fukuizumi, Representation formula for stochastic Schrödinger evolution equations and applications, Nonlinearity, 25 (2012), 2993-3022. doi: 10.1088/0951-7715/25/11/2993. [15] A. de Bouard and E. Gautier, Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise, Discrete. Contin. Dyn. Syst., 26 (2010), 857-871. doi: 10.3934/dcds.2010.26.857. [16] L. Di Menza, Numerical computation of solitons for optical systems, M2AN, Math. Model. Numer. Anal., 43 (2009), 173-208. doi: 10.1051/m2an:2008044. [17] L. Di Menza and F. Hadj Selem, Numerical study of solitons for nonlinear Schrödinger Equation with harmonic potential, preprint. [18] G. Fibich and N. Gavish, Theory of singular vortex solutions of the nonlinear Schrödinger equation, Physica D., 237 (2008), 2696-2730. doi: 10.1016/j.physd.2008.04.018. [19] M. E. Gehm, K. M. O'Hara, T. A. Savard and J. E. Thomas, Dynamics of noise-induced heating in atom traps, Phys. Rev. A, 58 (1998), 3914-3921. doi: 10.1103/PhysRevA.58.3914. [20] J. Ginibre and G. Velo, On the global Cauchy problem for some non linear Schrödinger equations, Ann. Inst. Henri Poincaré. Anal. Non linéaire, 1 (1984), 309-323. [21] J. Iaia and H. Warchall, Nonradial solutions of semilinear elliptic equation in two dimensions, J. Differential Equations, 119 (1995), 533-558. doi: 10.1006/jdeq.1995.1101. [22] J. Fröhlich, S. Gustafson, B. L. Jonsson and I. M. Sigal, Solitary wave dynamics in an external potential, Commun. Math. Phys., 250 (2004), 613-642. doi: 10.1007/s00220-004-1128-1. [23] B. L. Jonsson, J. Fröhlich, S. Gustafson and I. M. Sigal, Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré, 7 (2006), 621-660. doi: 10.1007/s00023-006-0263-y. [24] R. Kollár, Existence and Stability Of Vortex Solutions of Certain Nonlinear Schrödinger Equations, Ph.D. thesis, University of Maryland, College park, 2004. [25] R. Kollár and R. L. Pego, Spectral stability of vortices in two-dimensional Bose-Einstein condensates via the Evans function and Krein signature, Appl. Math. Res. Express. AMRX., 1 (2012), 1-46. [26] M. Maeda, Symmetry Breaking and Stability of Standing Waves of Nonlinear Schrödinger Equations, Master thesis, Kyoto University, 2008. [27] T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation, Diff. Integral Equations, 18 (2005), 431-450. [28] T. Mizumachi, Instability of vortex solitons for 2D focusing NLS, Adv. Diff. Equ., 12 (2007), 241-264. [29] Y. G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274. doi: 10.1016/0022-0396(89)90123-X. [30] R. L. Pego and H. Warchall, Spectrally stable encapsulated vortices for nonlinear Schrödinger equations, J. Nonlinear. Sci., 12 (2002), 347-394. doi: 10.1007/s00332-002-0475-3. [31] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford Univ. Press, New York, 2003. [32] M. Quiroga-Teixeiro and H. Michinel, Stable azimuthal stationary state in quintic nonlinear media, J. Opt. Soc. Am. B, 14 (1997), 2004-2009. doi: 10.1364/JOSAB.14.002004. [33] M. Reed and B. Simon, Method of Modern Mathematical Physics, I,II,III,IV Academic Press, 1975. [34] T. A. Savard, K. M. O'Hara and J. E. Thomas, Laser-noise induced heating in far-off resonance optical traps, Phys. Rev. A, 56 (1997), R1095-R1098. doi: 10.1103/PhysRevA.56.R1095. [35] R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys., 229 (2002), 491-509. doi: 10.1007/s00220-002-0695-2. [36] S. K. Suslov, Dynamical invariants for variable quadratic Hamiltonians, Phys. Scr., 81 (2010), 055006. doi: 10.1088/0031-8949/81/05/055006. [37] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, Second edition, Oxford at the Clarendon press, 1962. [38] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.

show all references

##### References:
 [1] F. Kh. Abdullaev, B. B. Baizakov and V. V. Konotop, Dynamics of a Bose-Einstein condensate in optical trap, in Nonlinearity and Disorder: Theory and Applications edited by F.Kh. Abdullaev, O. Bang and M.P. Soerensen, NATO Science Series, Kluwer Dodrecht, 45 (2001), 69-78. doi: 10.1007/978-94-010-0542-5_7. [2] F. Kh. Abdullaev, J. C. Bronski and G. Papanicolaou, Soliton perturbations and the random Kepler problem, Physica D, 135 (2000), 369-386. doi: 10.1016/S0167-2789(99)00118-9. [3] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. doi: 10.1137/S1064827503422956. [4] W. Bao and Y. Zhang, Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Models Methods Appl. Sci., 15 (2005), 1863-1896. doi: 10.1142/S021820250500100X. [5] F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer Academic Publishers, 1983. [6] C. C. Bradley, et al., Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75 (1995), 1687-1691. doi: 10.1103/PhysRevLett.75.1687. [7] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, Providence, RI: American Mathematical Society/Courant Institute of Mathematical Sciences, 2003. [8] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM. J. Math. Anal., 39 (2007/08), 1070-1111. doi: 10.1137/050648389. [9] K. B. Davis, et al., Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3974. doi: 10.1109/EQEC.1996.561567. [10] A. de Bouard and A. Debussche, Random modulation of solitons for the stochastic Korteweg-de Vries equation, Ann. I. H. Poincaré Anal. Non Linéaire, 24 (2007), 251-278. doi: 10.1016/j.anihpc.2006.03.009. [11] A. de Bouard and A. Debussche, Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise, Electron. J. Probab., 14 (2009), 1727-1744. doi: 10.1214/EJP.v14-683. [12] A. de Bouard and R. Fukuizumi, Stochastic fluctuations in the Gross-Pitaevskii equation, Nonlinearity, 20 (2007), 2823-2844. doi: 10.1088/0951-7715/20/12/005. [13] A. de Bouard and R. Fukuizumi, Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation, Asymptot. Anal., 63 (2009), 189-235. [14] A. de Bouard and R. Fukuizumi, Representation formula for stochastic Schrödinger evolution equations and applications, Nonlinearity, 25 (2012), 2993-3022. doi: 10.1088/0951-7715/25/11/2993. [15] A. de Bouard and E. Gautier, Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise, Discrete. Contin. Dyn. Syst., 26 (2010), 857-871. doi: 10.3934/dcds.2010.26.857. [16] L. Di Menza, Numerical computation of solitons for optical systems, M2AN, Math. Model. Numer. Anal., 43 (2009), 173-208. doi: 10.1051/m2an:2008044. [17] L. Di Menza and F. Hadj Selem, Numerical study of solitons for nonlinear Schrödinger Equation with harmonic potential, preprint. [18] G. Fibich and N. Gavish, Theory of singular vortex solutions of the nonlinear Schrödinger equation, Physica D., 237 (2008), 2696-2730. doi: 10.1016/j.physd.2008.04.018. [19] M. E. Gehm, K. M. O'Hara, T. A. Savard and J. E. Thomas, Dynamics of noise-induced heating in atom traps, Phys. Rev. A, 58 (1998), 3914-3921. doi: 10.1103/PhysRevA.58.3914. [20] J. Ginibre and G. Velo, On the global Cauchy problem for some non linear Schrödinger equations, Ann. Inst. Henri Poincaré. Anal. Non linéaire, 1 (1984), 309-323. [21] J. Iaia and H. Warchall, Nonradial solutions of semilinear elliptic equation in two dimensions, J. Differential Equations, 119 (1995), 533-558. doi: 10.1006/jdeq.1995.1101. [22] J. Fröhlich, S. Gustafson, B. L. Jonsson and I. M. Sigal, Solitary wave dynamics in an external potential, Commun. Math. Phys., 250 (2004), 613-642. doi: 10.1007/s00220-004-1128-1. [23] B. L. Jonsson, J. Fröhlich, S. Gustafson and I. M. Sigal, Long time motion of NLS solitary waves in a confining potential, Ann. Henri Poincaré, 7 (2006), 621-660. doi: 10.1007/s00023-006-0263-y. [24] R. Kollár, Existence and Stability Of Vortex Solutions of Certain Nonlinear Schrödinger Equations, Ph.D. thesis, University of Maryland, College park, 2004. [25] R. Kollár and R. L. Pego, Spectral stability of vortices in two-dimensional Bose-Einstein condensates via the Evans function and Krein signature, Appl. Math. Res. Express. AMRX., 1 (2012), 1-46. [26] M. Maeda, Symmetry Breaking and Stability of Standing Waves of Nonlinear Schrödinger Equations, Master thesis, Kyoto University, 2008. [27] T. Mizumachi, Vortex solitons for 2D focusing nonlinear Schrödinger equation, Diff. Integral Equations, 18 (2005), 431-450. [28] T. Mizumachi, Instability of vortex solitons for 2D focusing NLS, Adv. Diff. Equ., 12 (2007), 241-264. [29] Y. G. Oh, Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials, J. Differential Equations, 81 (1989), 255-274. doi: 10.1016/0022-0396(89)90123-X. [30] R. L. Pego and H. Warchall, Spectrally stable encapsulated vortices for nonlinear Schrödinger equations, J. Nonlinear. Sci., 12 (2002), 347-394. doi: 10.1007/s00332-002-0475-3. [31] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford Univ. Press, New York, 2003. [32] M. Quiroga-Teixeiro and H. Michinel, Stable azimuthal stationary state in quintic nonlinear media, J. Opt. Soc. Am. B, 14 (1997), 2004-2009. doi: 10.1364/JOSAB.14.002004. [33] M. Reed and B. Simon, Method of Modern Mathematical Physics, I,II,III,IV Academic Press, 1975. [34] T. A. Savard, K. M. O'Hara and J. E. Thomas, Laser-noise induced heating in far-off resonance optical traps, Phys. Rev. A, 56 (1997), R1095-R1098. doi: 10.1103/PhysRevA.56.R1095. [35] R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys., 229 (2002), 491-509. doi: 10.1007/s00220-002-0695-2. [36] S. K. Suslov, Dynamical invariants for variable quadratic Hamiltonians, Phys. Scr., 81 (2010), 055006. doi: 10.1088/0031-8949/81/05/055006. [37] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part I, Second edition, Oxford at the Clarendon press, 1962. [38] M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.
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