July  2019, 24(7): 3175-3193. doi: 10.3934/dcdsb.2018306

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2. 

College of Science, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Ruikuan Liu

Received  July 2018 Revised  July 2018 Published  October 2018

Fund Project: The work was supported by the Young Scholars Development Fund of SWPU (Grant No. 201899010079), and by the Natural Science Foundation of China (11771306).

The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

Citation: Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306
References:
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M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201.   Google Scholar

[2]

W. Bao and Y. Zhang, Dynamical laws of the coupled Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Methods Appl. Anal., 17 (2010), 49-80.  doi: 10.4310/MAA.2010.v17.n1.a2.  Google Scholar

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I. Bloch, M. Greiner, O. Mandel, T. W. Hänsch and T. Esslinger, Sympathetic cooling of 85 Rb, and 87 Rb, Phys. Rev. A, 64 (2001), p2. Google Scholar

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R. Coldea, D. A. Tennant, E. M. Wheeler et al., Quantum criticality in an ising chain: Experimental evidence for emergent e8 symmetry, Science, 327 (2010), 177-180. Google Scholar

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F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev.Mod. Phys., 71 (1998), 463-512.   Google Scholar

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K. B. DavisM. O. MewesM. R. AndrewsN. J. van DrutenD. S. DurfeeD. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 39-69.   Google Scholar

[10]

F. J. DysonE. H. Lieb and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys., 18 (1978), 335-383.  doi: 10.1007/BF01106729.  Google Scholar

[11]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys., 81 (2009), 647.   Google Scholar

[12]

C. Hsia, C. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A. 471 (2015), 20140353, 24 pp. doi: 10.1098/rspa.2014.0353.  Google Scholar

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M. Greiner, I. Bloch and T. W. Hänsch, et. al., Magnetic transport of trapped cold atoms over a large distance, Physical Review A, 63 (2001), 031401. Google Scholar

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T.-L. Ho, Spinor bose condensates in optical traps, Physical review letters, 81 (1998), 742-745.   Google Scholar

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J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics, 6 (1973), 1181.   Google Scholar

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L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Butterworth-Heinemann, Amsterdam, 2003. Google Scholar

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H. LiuT. SengulS. Wang and P. Zhang, Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.  doi: 10.4310/CMS.2015.v13.n5.a10.  Google Scholar

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T. Ma, R. Liu and J. Yang, Physical World from the Mathematical Point of View: Statistical Physics and Critical Phase Transition Theory (in Chinese), Science Press, Beijing, 2018. Google Scholar

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T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd. : Hackensack, NJ, 2005. doi: 10.1142/5827.  Google Scholar

[24]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[25]

T. Ma and S. Wang, Dynamic transition and pattern formation in Taylor problem, Chin. Ann. Math. Ser. B, 31 (2010), 953-974.  doi: 10.1007/s11401-010-0610-7.  Google Scholar

[26]

T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446.  Google Scholar

[27]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.  Google Scholar

[28]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[29]

T. Ma and S. Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 2015. Google Scholar

[30]

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[31]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822-1825.   Google Scholar

[32]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.   Google Scholar

[33]

J. Rogel-Salazar, The Gross-Pitaevskii equation and Bose-Einstein condensates, Eur. J. Phys., 34 (2013), 247-257.   Google Scholar

[34]

R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys., 229 (2002), 491-509.  doi: 10.1007/s00220-002-0695-2.  Google Scholar

[35]

R. Seiringer, Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation, J. Math. Phys., 55 (2014), 075209, 18pp. doi: 10.1063/1.4881536.  Google Scholar

[36]

T. SengulJ. Shen and S. Wang, Pattern formations of 2D Rayleigh-Bénard convection with no-slip boundary conditions for the velocity at the critical length scales, Math. Methods Appl. Sci., 38 (2015), 3792-3806.  doi: 10.1002/mma.3317.  Google Scholar

[37]

T. Sengul and S. Wang, Pattern formation in Rayleigh-Bénard convection, Commun. Math. Sci., 11 (2013), 315-343.  doi: 10.4310/CMS.2013.v11.n1.a10.  Google Scholar

[38]

A. Sütő, Equivalence of Bose-Einstein condensation and symmetry breaking, Phys. Rev. Lett., 94 (2005), 080402. Google Scholar

[39]

M. Vojta, Quantum phase transition, Rep. Prog. Phys., 66 (2003), 2069-2110.  doi: 10.1088/0034-4885/66/12/R01.  Google Scholar

show all references

References:
[1]

M. H. AndersonJ. R. EnsherM. R. MatthewsC. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201.   Google Scholar

[2]

W. Bao and Y. Zhang, Dynamical laws of the coupled Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates, Methods Appl. Anal., 17 (2010), 49-80.  doi: 10.4310/MAA.2010.v17.n1.a2.  Google Scholar

[3]

D. BitkoT. F. Rosenbaum and G. Aeppli, Quantum Critical Behavior for a Model Magnet, Phys. Rev. Lett., 77 (1996), 940.   Google Scholar

[4]

I. Bloch, M. Greiner, O. Mandel, T. W. Hänsch and T. Esslinger, Sympathetic cooling of 85 Rb, and 87 Rb, Phys. Rev. A, 64 (2001), p2. Google Scholar

[5]

P. Chandra, G. G. Lonzarich, S. E. Rowley and J. F. Scott, Prospects and applications near ferroelectric quantum phase transitions: A key issues review, Rep. Progr. Phys., 80 (2017), 112502, 24pp. doi: 10.1088/1361-6633/aa82d2.  Google Scholar

[6]

L. W. ClarkL. Feng and C. Chin, Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition, Science, 354 (2016), 606-610.  doi: 10.1126/science.aaf9657.  Google Scholar

[7]

R. Coldea, D. A. Tennant, E. M. Wheeler et al., Quantum criticality in an ising chain: Experimental evidence for emergent e8 symmetry, Science, 327 (2010), 177-180. Google Scholar

[8]

F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev.Mod. Phys., 71 (1998), 463-512.   Google Scholar

[9]

K. B. DavisM. O. MewesM. R. AndrewsN. J. van DrutenD. S. DurfeeD. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 39-69.   Google Scholar

[10]

F. J. DysonE. H. Lieb and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys., 18 (1978), 335-383.  doi: 10.1007/BF01106729.  Google Scholar

[11]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys., 81 (2009), 647.   Google Scholar

[12]

C. Hsia, C. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A. 471 (2015), 20140353, 24 pp. doi: 10.1098/rspa.2014.0353.  Google Scholar

[13]

M. Greiner, I. Bloch and T. W. Hänsch, et. al., Magnetic transport of trapped cold atoms over a large distance, Physical Review A, 63 (2001), 031401. Google Scholar

[14]

T.-L. Ho, Spinor bose condensates in optical traps, Physical review letters, 81 (1998), 742-745.   Google Scholar

[15]

J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics, 6 (1973), 1181.   Google Scholar

[16]

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Butterworth-Heinemann, Amsterdam, 2003. Google Scholar

[17]

R. Liu, T. Ma, S. Wang and J. Yang, hermodynamical potentials of classical and quantum systems, Discrete Contin. Dyn. Syst. Ser. B., 2018, to appear. Google Scholar

[18]

H. LiuT. SengulS. Wang and P. Zhang, Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.  doi: 10.4310/CMS.2015.v13.n5.a10.  Google Scholar

[19]

E. K. Luckins and R. A. Van Gorder, Bose-Einstein condensation under the cubic-quintic Gross-Pitaevskii equation in radial domains, Ann. Physics, 388 (2018), 206-234.  doi: 10.1016/j.aop.2017.11.009.  Google Scholar

[20]

X. LuoY. ZouL. WuQ. LiuM. HanM. Tey and L. You, Deterministic entanglement generation from driving through quantum phase transitions, Science, 355 (2017), 620-623.  doi: 10.1126/science.aag1106.  Google Scholar

[21]

T. Ma, D. Li, R. Liu and J. Yang, Mathematical Theory for Quantum Phase Transitions, 2016, see arXiv: 1610.06988 Google Scholar

[22]

T. Ma, R. Liu and J. Yang, Physical World from the Mathematical Point of View: Statistical Physics and Critical Phase Transition Theory (in Chinese), Science Press, Beijing, 2018. Google Scholar

[23]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Publishing Co. Pte. Ltd. : Hackensack, NJ, 2005. doi: 10.1142/5827.  Google Scholar

[24]

T. Ma and S. Wang, Dynamic transition theory for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.  Google Scholar

[25]

T. Ma and S. Wang, Dynamic transition and pattern formation in Taylor problem, Chin. Ann. Math. Ser. B, 31 (2010), 953-974.  doi: 10.1007/s11401-010-0610-7.  Google Scholar

[26]

T. Ma and S. Wang, Phase transitions for Belousov-Zhabotinsky reactions, Math. Methods Appl. Sci., 34 (2011), 1381-1397.  doi: 10.1002/mma.1446.  Google Scholar

[27]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.  Google Scholar

[28]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[29]

T. Ma and S. Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 2015. Google Scholar

[30]

T. Ma and S. Wang, Topological Phase Transitions Ⅰ: Quantum Phase Transitions, to appear, (2018) or (Tian Ma, Shouhong Wang. Topological Phase Transitions Ⅰ: Quantum Phase Transitions, 2017. < hal-01651908 > ). Google Scholar

[31]

T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822-1825.   Google Scholar

[32]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.   Google Scholar

[33]

J. Rogel-Salazar, The Gross-Pitaevskii equation and Bose-Einstein condensates, Eur. J. Phys., 34 (2013), 247-257.   Google Scholar

[34]

R. Seiringer, Gross-Pitaevskii theory of the rotating Bose gas, Commun. Math. Phys., 229 (2002), 491-509.  doi: 10.1007/s00220-002-0695-2.  Google Scholar

[35]

R. Seiringer, Bose gases, Bose-Einstein condensation, and the Bogoliubov approximation, J. Math. Phys., 55 (2014), 075209, 18pp. doi: 10.1063/1.4881536.  Google Scholar

[36]

T. SengulJ. Shen and S. Wang, Pattern formations of 2D Rayleigh-Bénard convection with no-slip boundary conditions for the velocity at the critical length scales, Math. Methods Appl. Sci., 38 (2015), 3792-3806.  doi: 10.1002/mma.3317.  Google Scholar

[37]

T. Sengul and S. Wang, Pattern formation in Rayleigh-Bénard convection, Commun. Math. Sci., 11 (2013), 315-343.  doi: 10.4310/CMS.2013.v11.n1.a10.  Google Scholar

[38]

A. Sütő, Equivalence of Bose-Einstein condensation and symmetry breaking, Phys. Rev. Lett., 94 (2005), 080402. Google Scholar

[39]

M. Vojta, Quantum phase transition, Rep. Prog. Phys., 66 (2003), 2069-2110.  doi: 10.1088/0034-4885/66/12/R01.  Google Scholar

Figure 1.  The graph of the part of S2
Figure 2.  The graph of S1
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