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 (): 1070.  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 (): 1070.  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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