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2011, 10(6): 1589-1615. doi: 10.3934/cpaa.2011.10.1589

Vortex interaction dynamics in trapped Bose-Einstein condensates

1. 

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada Campus de Fuentenueva s/n, 18071 Granada, Spain

2. 

Nonlinear Physics Group, Departamento de Física Aplicada I, Universidad de Sevilla, 41012 Sevilla, Spain

3. 

Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany, Germany

4. 

Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece

5. 

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9315, United States

Received  July 2010 Revised  May 2011 Published  May 2011

Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates (BECs), we illustrate that such systems can be accurately described by ordinary differential equations (ODEs) incorporating the precession and interaction dynamics of vortices in harmonic traps. This dynamics is tackled in detail at the ODE level, both for the simpler case of equal charge vortices, and for the more complicated (yet also experimentally relevant) case of opposite charge vortices. In the former case, we identify the dynamics as being chiefly quasi-periodic (although potentially periodic), while in the latter, irregular dynamics may ensue when suitable external drive of the BEC cloud is also considered. Our analytical findings are corroborated by numerical computations of the reduced ODE system.
Citation: Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589
References:
[1]

L. M. Pismen, "Vortices in Nonlinear Fields,", Oxford Science Publications, (1999).

[2]

A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics,", Springer-Verlag, (1993).

[3]

Yu. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies and L. M. Pismen, Dynamics of optical vortex solitons,, Opt. Commun., 152 (1998), 198. doi: 10.1016/S0030-4018(98)00149-7.

[4]

A. Dreischuh, S. Chevrenkov, D. Neshev, G. G. Paulus and H. Walther, Generation of lattice structures of optical vortices,, J. Opt. Soc. Am. B, 19 (2002), 550. doi: 10.1364/JOSAB.19.000550.

[5]

A. S. Desyatnikov, Yu. S. Kivshar and L. Torner, Optical vortices and vortex solitons,, Prog. Optics, 47 (2005), 291. doi: 10.1016/S0079-6638(05)47006-7.

[6]

L. P. Pitaevskii and S. Stringari, "Bose-Einstein Condensation,", Oxford University Press, (2003).

[7]

C. J. Pethick and H. Smith, "Bose-Einstein Condensation in Dilute Gases,", Cambridge University Press, (2002).

[8]

P. G. Kevrekidis, D. J. Frantzeskakis and R. Carretero-González, "Emergent Nonlinear Phenomena in Bose-Einstein Condensates. Theory and Experiment,", Springer-Verlag, (2008). doi: 10.1007/978-3-540-73591-5_1.

[9]

A. L. Fetter and A. A. Svidzinksy, Vortices in a trapped dilute Bose-Einstein condensate,, J. Phys.: Cond. Matt., 13 (2001). doi: 10.1088/0953-8984/13/12/201.

[10]

P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis and I. G. Kevrekidis, Vortices in Bose-Einstein condensates: some recent developments,, Mod. Phys. Lett. B, 18 (2004), 1481. doi: 10.1142/S0217984904007967.

[11]

P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective,, SIAM Rev., 51 (2009), 501. doi: 10.1137/07068597X.

[12]

R. Carretero-González, P. G. Kevrekidis and D. J. Frantzeskakis, Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/7/R01.

[13]

A. L. Fetter, Rotating trapped Bose-Einstein condensates,, Rev. Mod. Phys., 81 (2009), 647. doi: 10.1103/RevModPhys.81.647.

[14]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, Vortices in a Bose-Einstein condensate,, Phys. Rev. Lett., 83 (1999), 2498. doi: 10.1103/PhysRevLett.83.2498.

[15]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation,, Phys. Rev. Lett., 86 (2001), 4443. doi: 10.1103/PhysRevLett.86.4443.

[16]

K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate,, Phys. Rev. Lett., 84 (2000), 806. doi: 10.1103/PhysRevLett.84.806.

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S. Sinha and Y. Castin, Dynamic instability of a Rotating Bose-Einstein condensate,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.190402.

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C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a Stirred Bose-Einstein condensate,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.210402.

[20]

D. R. Scherer, C. N. Weiler, T. W. Neely and B. P. Anderson, Vortex formation by merging of multiple trapped Bose-Einstein condensates,, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.110402.

[21]

R. Carretero-González, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis, Vortex structures formed by the interference of sliced condensates,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.023605.

[22]

R. Carretero-González, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis and C. N. Weiler, Dynamics of vortex formation in merging Bose-Einstein condensate fragments,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.033625.

[23]

G. Ruben, D. M. Paganin and M. J. Morgan, Vortex-lattice formation and melting in a nonrotating Bose-Einstein condensate,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.013631.

[24]

C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis and B. P. Anderson, Spontaneous vortices in the formation of Bose-Einstein condensates,, Nature, 455 (2008), 948. doi: 10.1038/nature07334.

[25]

A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard and W. Ketterle, Imprinting vortices in a Bose-Einstein condensate using topological phases,, Phys. Rev. Lett., 89 (2002). doi: 10.1103/PhysRevLett.89.190403.

[26]

Y. Shin, M. Saba, M. Vengalattore, T. A. Pasquini, C. Sanner, A. E. Leanhardt, M. Prentiss, D. E. Pritchard and W. Ketterle, Dynamical instability of a doubly quantized vortex in a Bose-Einstein condensate,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.160406.

[27]

T. Isoshima, M. Okano, H. Yasuda, K. Kasa, J. A. M. Huhtamäki, M. Kumakura and Y. Takahashi, Spontaneous splitting of a quadruply charged vortex,, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.200403.

[28]

L.-C. Crasovan, V. Vekslerchik, V. M. Pérez-García, J. P. Torres, D. Mihalache and L. Torner, Stable vortex dipoles in nonrotating Bose-Einstein condensates,, Phys. Rev. A, 68 (2003). doi: 10.1103/PhysRevA.68.063609.

[29]

M. Möttönen, S. M. M. Virtanen, T. Isoshima and M. M. Salomaa, Stationary vortex clusters in nonrotating Bose-Einstein condensates,, Phys. Rev. A, 71 (2005). doi: 10.1103/PhysRevA.71.033626.

[30]

V. Pietilä, M. Möttönen, T. Isoshima, J. A. M. Huhtamäki and S. M. M. Virtanen, Stability and dynamics of vortex clusters in nonrotated Bose-Einstein condensates,, Phys. Rev. A, 74 (2006). doi: 10.1103/PhysRevA.74.023603.

[31]

A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates,, Phys. Rev. A, 76 (2007). doi: 10.1103/PhysRevA.76.043602.

[32]

W. Li, M. Haque and S. Komineas, Vortex dipole in a trapped two-dimensional Bose-Einstein condensate,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.053610.

[33]

J.-P. Martikainen, K.-A. Suominen, L. Santos, T. Schulte and A. Sanpera, Generation and evolution of vortex-antivortex pairs in Bose-Einstein condensates,, Phys. Rev. A, 64 (2001). doi: 10.1103/PhysRevA.64.063602.

[34]

T. Schulte, L. Santos, A. Sanpera and M. Lewenstein, Vortex-vortex interactions in toroidally trapped Bose-Einstein condensates,, Phys. Rev. A, 66 (2002). doi: 10.1103/PhysRevA.66.033602.

[35]

S. McEndoo and Th. Busch, Small numbers of vortices in anisotropic traps,, Phys. Rev. A, 79 (2009). doi: 10.1103/PhysRevA.79.053616.

[36]

T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson, Observation of vortex dipoles in an oblate Bose-Einstein condensate,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.160401.

[37]

J. A. Seman, E. A. L. Henn, M. Haque, R. F. Shiozaki, E. R. F. Ramos, M. Caracanhas, P. Castilho, C. Castelo Branco, K. M. F. Magalhães and V. S. Bagnato, Three-vortex configurations in trapped Bose-Einstein condensates,, Phys. Rev. A, 82 (2010). doi: 10.1103/PhysRevA.82.033616.

[38]

D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin and D. S. Hall, Real-time dynamics of single vortex lines and vortex dipoles in a Bose-Einstein condensate,, Science, 329 (2010), 1182. doi: 10.1126/science.1191224.

[39]

S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González and P. Schmelcher, Bifurcations, stability and dynamics of multiple matter-wave vortex states,, Phys. Rev. A, 82 (2010). doi: 10.1103/PhysRevA.82.013646.

[40]

B. Jackson, J. F. McCann and C. S. Adams, Vortex line and ring dynamics in trapped Bose-Einstein condensates,, Phys. Rev. A, 61 (1999). doi: 10.1103/PhysRevA.61.013604.

[41]

A. A. Svidzinsky and A. L. Fetter, Stability of a vortex in a trapped Bose-Einstein condensate,, Phys. Rev. Lett., 84 (2000), 5919. doi: 10.1103/PhysRevLett.84.5919.

[42]

J. Tempere and J. T. Devreese, Vortex dynamics in a parabolically confined Bose-Einstein condensate,, Solid State Comm., 113 (2000), 471. doi: 10.1016/S0038-1098(99)00495-0.

[43]

B. P. Anderson, P. C. Haljan, C. E. Wieman and E. A. Cornell, Vortex precession in Bose-Einstein condensates: observations with filled and empty cores,, Phys. Rev. Lett., 85 (2000), 2857. doi: 10.1103/PhysRevLett.85.2857.

[44]

S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González and P. Schmelcher, Stability and dynamics of matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations,, J. Phys. B: At. Mo. Opt. Phys., 43 (2010). doi: 10.1088/0953-4075/43/15/155303.

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A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems,", Birkh\, (1993).

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A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach,, J. Differential Equations, 244 (2008), 3235. doi: 10.1016/j.jde.2007.11.005.

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A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: the repulsive case,, To appear in Adv. Nonlinear Stud., (2011).

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A. Fonda and A. J. Ureña, Periodic, subharmonic and quasi-periodic oscillations under the action of a central force,, Discrete Cont. Dyn. Syst. A, 29 (2011), 169. doi: 10.3934/dcds.2011.29.169.

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P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,, J. Diff. Eq., 190 (2003), 643. doi: 10.1016/S0022-0396(02)00152-3.

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show all references

References:
[1]

L. M. Pismen, "Vortices in Nonlinear Fields,", Oxford Science Publications, (1999).

[2]

A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics,", Springer-Verlag, (1993).

[3]

Yu. S. Kivshar, J. Christou, V. Tikhonenko, B. Luther-Davies and L. M. Pismen, Dynamics of optical vortex solitons,, Opt. Commun., 152 (1998), 198. doi: 10.1016/S0030-4018(98)00149-7.

[4]

A. Dreischuh, S. Chevrenkov, D. Neshev, G. G. Paulus and H. Walther, Generation of lattice structures of optical vortices,, J. Opt. Soc. Am. B, 19 (2002), 550. doi: 10.1364/JOSAB.19.000550.

[5]

A. S. Desyatnikov, Yu. S. Kivshar and L. Torner, Optical vortices and vortex solitons,, Prog. Optics, 47 (2005), 291. doi: 10.1016/S0079-6638(05)47006-7.

[6]

L. P. Pitaevskii and S. Stringari, "Bose-Einstein Condensation,", Oxford University Press, (2003).

[7]

C. J. Pethick and H. Smith, "Bose-Einstein Condensation in Dilute Gases,", Cambridge University Press, (2002).

[8]

P. G. Kevrekidis, D. J. Frantzeskakis and R. Carretero-González, "Emergent Nonlinear Phenomena in Bose-Einstein Condensates. Theory and Experiment,", Springer-Verlag, (2008). doi: 10.1007/978-3-540-73591-5_1.

[9]

A. L. Fetter and A. A. Svidzinksy, Vortices in a trapped dilute Bose-Einstein condensate,, J. Phys.: Cond. Matt., 13 (2001). doi: 10.1088/0953-8984/13/12/201.

[10]

P. G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis and I. G. Kevrekidis, Vortices in Bose-Einstein condensates: some recent developments,, Mod. Phys. Lett. B, 18 (2004), 1481. doi: 10.1142/S0217984904007967.

[11]

P. K. Newton and G. Chamoun, Vortex lattice theory: A particle interaction perspective,, SIAM Rev., 51 (2009), 501. doi: 10.1137/07068597X.

[12]

R. Carretero-González, P. G. Kevrekidis and D. J. Frantzeskakis, Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/7/R01.

[13]

A. L. Fetter, Rotating trapped Bose-Einstein condensates,, Rev. Mod. Phys., 81 (2009), 647. doi: 10.1103/RevModPhys.81.647.

[14]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, Vortices in a Bose-Einstein condensate,, Phys. Rev. Lett., 83 (1999), 2498. doi: 10.1103/PhysRevLett.83.2498.

[15]

K. W. Madison, F. Chevy, V. Bretin and J. Dalibard, Stationary states of a rotating Bose-Einstein condensate: routes to vortex nucleation,, Phys. Rev. Lett., 86 (2001), 4443. doi: 10.1103/PhysRevLett.86.4443.

[16]

K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose-Einstein condensate,, Phys. Rev. Lett., 84 (2000), 806. doi: 10.1103/PhysRevLett.84.806.

[17]

A. Recati, F. Zambelli and S. Stringari, Overcritical rotation of a trapped Bose-Einstein condensate,, Phys. Rev. Lett., 86 (2001), 377. doi: 10.1103/PhysRevLett.86.377.

[18]

S. Sinha and Y. Castin, Dynamic instability of a Rotating Bose-Einstein condensate,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.190402.

[19]

C. Raman, J. R. Abo-Shaeer, J. M. Vogels, K. Xu and W. Ketterle, Vortex nucleation in a Stirred Bose-Einstein condensate,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.210402.

[20]

D. R. Scherer, C. N. Weiler, T. W. Neely and B. P. Anderson, Vortex formation by merging of multiple trapped Bose-Einstein condensates,, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.110402.

[21]

R. Carretero-González, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis, Vortex structures formed by the interference of sliced condensates,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.023605.

[22]

R. Carretero-González, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis and C. N. Weiler, Dynamics of vortex formation in merging Bose-Einstein condensate fragments,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.033625.

[23]

G. Ruben, D. M. Paganin and M. J. Morgan, Vortex-lattice formation and melting in a nonrotating Bose-Einstein condensate,, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.013631.

[24]

C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis and B. P. Anderson, Spontaneous vortices in the formation of Bose-Einstein condensates,, Nature, 455 (2008), 948. doi: 10.1038/nature07334.

[25]

A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard and W. Ketterle, Imprinting vortices in a Bose-Einstein condensate using topological phases,, Phys. Rev. Lett., 89 (2002). doi: 10.1103/PhysRevLett.89.190403.

[26]

Y. Shin, M. Saba, M. Vengalattore, T. A. Pasquini, C. Sanner, A. E. Leanhardt, M. Prentiss, D. E. Pritchard and W. Ketterle, Dynamical instability of a doubly quantized vortex in a Bose-Einstein condensate,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.160406.

[27]

T. Isoshima, M. Okano, H. Yasuda, K. Kasa, J. A. M. Huhtamäki, M. Kumakura and Y. Takahashi, Spontaneous splitting of a quadruply charged vortex,, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.200403.

[28]

L.-C. Crasovan, V. Vekslerchik, V. M. Pérez-García, J. P. Torres, D. Mihalache and L. Torner, Stable vortex dipoles in nonrotating Bose-Einstein condensates,, Phys. Rev. A, 68 (2003). doi: 10.1103/PhysRevA.68.063609.

[29]

M. Möttönen, S. M. M. Virtanen, T. Isoshima and M. M. Salomaa, Stationary vortex clusters in nonrotating Bose-Einstein condensates,, Phys. Rev. A, 71 (2005). doi: 10.1103/PhysRevA.71.033626.

[30]

V. Pietilä, M. Möttönen, T. Isoshima, J. A. M. Huhtamäki and S. M. M. Virtanen, Stability and dynamics of vortex clusters in nonrotated Bose-Einstein condensates,, Phys. Rev. A, 74 (2006). doi: 10.1103/PhysRevA.74.023603.

[31]

A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates,, Phys. Rev. A, 76 (2007). doi: 10.1103/PhysRevA.76.043602.

[32]

W. Li, M. Haque and S. Komineas, Vortex dipole in a trapped two-dimensional Bose-Einstein condensate,, Phys. Rev. A, 77 (2008). doi: 10.1103/PhysRevA.77.053610.

[33]

J.-P. Martikainen, K.-A. Suominen, L. Santos, T. Schulte and A. Sanpera, Generation and evolution of vortex-antivortex pairs in Bose-Einstein condensates,, Phys. Rev. A, 64 (2001). doi: 10.1103/PhysRevA.64.063602.

[34]

T. Schulte, L. Santos, A. Sanpera and M. Lewenstein, Vortex-vortex interactions in toroidally trapped Bose-Einstein condensates,, Phys. Rev. A, 66 (2002). doi: 10.1103/PhysRevA.66.033602.

[35]

S. McEndoo and Th. Busch, Small numbers of vortices in anisotropic traps,, Phys. Rev. A, 79 (2009). doi: 10.1103/PhysRevA.79.053616.

[36]

T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson, Observation of vortex dipoles in an oblate Bose-Einstein condensate,, Phys. Rev. Lett., 104 (2010). doi: 10.1103/PhysRevLett.104.160401.

[37]

J. A. Seman, E. A. L. Henn, M. Haque, R. F. Shiozaki, E. R. F. Ramos, M. Caracanhas, P. Castilho, C. Castelo Branco, K. M. F. Magalhães and V. S. Bagnato, Three-vortex configurations in trapped Bose-Einstein condensates,, Phys. Rev. A, 82 (2010). doi: 10.1103/PhysRevA.82.033616.

[38]

D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin and D. S. Hall, Real-time dynamics of single vortex lines and vortex dipoles in a Bose-Einstein condensate,, Science, 329 (2010), 1182. doi: 10.1126/science.1191224.

[39]

S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González and P. Schmelcher, Bifurcations, stability and dynamics of multiple matter-wave vortex states,, Phys. Rev. A, 82 (2010). doi: 10.1103/PhysRevA.82.013646.

[40]

B. Jackson, J. F. McCann and C. S. Adams, Vortex line and ring dynamics in trapped Bose-Einstein condensates,, Phys. Rev. A, 61 (1999). doi: 10.1103/PhysRevA.61.013604.

[41]

A. A. Svidzinsky and A. L. Fetter, Stability of a vortex in a trapped Bose-Einstein condensate,, Phys. Rev. Lett., 84 (2000), 5919. doi: 10.1103/PhysRevLett.84.5919.

[42]

J. Tempere and J. T. Devreese, Vortex dynamics in a parabolically confined Bose-Einstein condensate,, Solid State Comm., 113 (2000), 471. doi: 10.1016/S0038-1098(99)00495-0.

[43]

B. P. Anderson, P. C. Haljan, C. E. Wieman and E. A. Cornell, Vortex precession in Bose-Einstein condensates: observations with filled and empty cores,, Phys. Rev. Lett., 85 (2000), 2857. doi: 10.1103/PhysRevLett.85.2857.

[44]

S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R. Carretero-González and P. Schmelcher, Stability and dynamics of matter-wave vortices in the presence of collisional inhomogeneities and dissipative perturbations,, J. Phys. B: At. Mo. Opt. Phys., 43 (2010). doi: 10.1088/0953-4075/43/15/155303.

[45]

A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems,", Birkh\, (1993).

[46]

A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach,, J. Differential Equations, 244 (2008), 3235. doi: 10.1016/j.jde.2007.11.005.

[47]

A. Fonda and R. Toader, Periodic orbits of radially symmetric systems with a singularity: the repulsive case,, To appear in Adv. Nonlinear Stud., (2011).

[48]

A. Fonda and A. J. Ureña, Periodic, subharmonic and quasi-periodic oscillations under the action of a central force,, Discrete Cont. Dyn. Syst. A, 29 (2011), 169. doi: 10.3934/dcds.2011.29.169.

[49]

P. J. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,, J. Diff. Eq., 190 (2003), 643. doi: 10.1016/S0022-0396(02)00152-3.

[50]

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