High-order numerical methods with mass and energy conservation for spin–orbit-coupled Bose–Einstein condensates

References

• M.H. Anderson, J.R. Ensher, M.R. Matthewa, C.E. Wieman, and E.A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269 (1995), pp. 198–201. doi: 10.1126/science.269.5221.198
   PubMed Web of Science ®Google Scholar
• X. Antoine, W. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun. 184 (2013), pp. 2621–2633. doi: 10.1016/j.cpc.2013.07.012
   Web of Science ®Google Scholar
• W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic Related Models. 6 (2013), pp. 1–135. doi: 10.3934/krm.2013.6.1
   Web of Science ®Google Scholar
• W. Bao and Y. Cai, Ground states and dynamics of spin-orbit-coupled Bose-Einstein condensates, SIAM J. Appl. Math. 75 (2015), pp. 492–517. doi: 10.1137/140979241
   Web of Science ®Google Scholar
• B.A. Bernevig, T.L. Hughes, and S.C. Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 314 (2006), pp. 1757–1761. doi: 10.1126/science.1133734
   PubMed Web of Science ®Google Scholar
• Q. Chang, E. Jia, and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys. 148 (1999), pp. 397–415. doi: 10.1006/jcph.1998.6120
   Web of Science ®Google Scholar
• L.W. Cheuk, A.T. Sommer, Z. Hadzibabic, T. Yefsah, W.S. Bakr, and M.W. Zwierlein, Spin-injection spectroscopy of a spin-orbit coupled Fermi gas, Phys. Rev. Lett. 109 (2012), p. 095302. doi: 10.1103/PhysRevLett.109.095302
   PubMed Web of Science ®Google Scholar
• M. Dehghan and A. Taleei, A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Commun. 181 (2010), pp. 43–51. doi: 10.1016/j.cpc.2009.08.015
   Web of Science ®Google Scholar
• Y. Deng, J. Cheng, H. Jing, C.P. Sun, and S. Yi, Spin-orbit-coupled dipolar Bose-Einstein condensates, Phys. Rev. Lett. 108 (2012), pp. 125301.
   PubMed Web of Science ®Google Scholar
• L.R.T. Gardner, G.A. Gardner, S.I. Zaki, and Z. El Sahrawi, B-spline finite element studies of the non-linear Schrödinger equation, Comput. Method. Appl. M. 108 (1993), pp. 303–318. doi: 10.1016/0045-7825(93)90007-K
   Web of Science ®Google Scholar
• M.Z. Hasan and C.L. Kane, Colloquium: topological insulators, Rev. Mod. Phys. 82 (2010), pp. 3045–3067. doi: 10.1103/RevModPhys.82.3045
   Web of Science ®Google Scholar
• D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, and M.Z. Hasan, A topological Dirac insulator in a quantum spin Hall phase, Nature 452 (2008), pp. 970–974. doi: 10.1038/nature06843
   PubMed Web of Science ®Google Scholar
• H. Hu, B. Ramachandhran, H. Pu, and X.J. Liu, Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps, Phys. Rev. Lett. 108 (2012), pp. 010402.
   PubMed Web of Science ®Google Scholar
• A. Jacob, P. Öhberg, J.G. Juzeliūnas, and L. Santos, Landau levels of cold atoms in non-Abelian gauge fields, New J. Phys. 10 (2008), p. 045022. doi: 10.1088/1367-2630/10/4/045022
   Web of Science ®Google Scholar
• M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.L. Qi, and S.C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells, Science 318 (2007), pp. 766–770. doi: 10.1126/science.1148047
   PubMed Web of Science ®Google Scholar
• O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method, SIAM J. Numer. Anal. 36 (1998), pp. 1779–1807. doi: 10.1137/S0036142997330111
   Web of Science ®Google Scholar
• O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comp. 67 (1998), pp. 479–499. doi: 10.1090/S0025-5718-98-00946-6
   Web of Science ®Google Scholar
• Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Observation of the spin Hall effect in semiconductors, Science 306 (2004), pp. 1910–1913. doi: 10.1126/science.1105514
   PubMed Web of Science ®Google Scholar
• X.G. Li, Y. Cai, P. Wang, Operator-compensation methods with mass and energy conservation for solving the Gross-Pitaevskii equation. Appl. Numer. Math. 151 (2020), pp. 337–353. doi: 10.1016/j.apnum.2020.01.004
   Web of Science ®Google Scholar
• X.G. Li, C.K. Chan, and Y. Hou, A numerical method with particle conservation for the Maxwell-Dirac system, Appl. Math. Comput. 216 (2010), pp. 1096–1108.
   Web of Science ®Google Scholar
• S.C. Li and X.G. Li, High-order compact methods for the nonlinear Dirac equation, Comput. Appl. Math. 37 (2018), pp. 6483–6498. doi: 10.1007/s40314-018-0705-4
   Google Scholar
• S.C. Li and X.G. Li, High-order conservative schemes for the nonlinear Dirac equation, Int. J. Comput. Math. 2019. Published online.
   PubMed Web of Science ®Google Scholar
• X.G. Li, J. Zhu, R.P. Zhang, and S. Cao, A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation, J. Comput. Phys. 275 (2014), pp. 363–376. doi: 10.1016/j.jcp.2014.07.013
   Web of Science ®Google Scholar
• Y.J. Lin, R.L. Compton, K. Jiménez-García, and P.J.V. I.B. Spielman, Synthetic magnetic fields for ultracold neutral atoms, Nature 462 (2009), pp. 628–632. doi: 10.1038/nature08609
   PubMed Web of Science ®Google Scholar
• Y.J. Lin, R.L. Compton, K. Jiménez-García, W.D. Phillips, J.V. Porto, and I.B. Spielman, A synthetic electric force acting on neutral atoms, Nat. Phys. 7 (2011), pp. 531–534. doi: 10.1038/nphys1954
   Web of Science ®Google Scholar
• Y.J. Lin, K. Jiménez-García, and I.B. Spielman, Spin-orbit-coupled Bose-Einstein condensates, Nature 471 (2011), pp. 83–86. doi: 10.1038/nature09887
   PubMed Web of Science ®Google Scholar
• C.F. Liu and W.M. Liu, Spin-orbit-coupling-induced half-skyrmion excitations in rotating and rapidly quenched spin-1 Bose-Einstein condensates, Phys. Rev. A 86 (2012), p. 033602.
   Web of Science ®Google Scholar
• B. Ramachandhran, B. Opanchuk, X.J. Liu, H. Pu, P.D. Drummond, and H. Hu, Half-quantum vortex state in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 85 (2012), p. 023606. doi: 10.1103/PhysRevA.85.023606
   Google Scholar
• S. Sinha, R. Nath, and L. Santos, Trapped two-dimensional condensates with synthetic spin-orbit coupling, Phys. Rev. Lett. 107 (2011), p. 270401.
   PubMed Web of Science ®Google Scholar
• T.R. Taha, A numerical scheme for the nonlinear Schrödinger equation, Comput. Math. Appl. 22 (1991), pp. 77–84. doi: 10.1016/0898-1221(91)90208-L
   Web of Science ®Google Scholar
• M. Thalhammer, M. Caliari, and C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys. 228 (2009), pp. 822–832. doi: 10.1016/j.jcp.2008.10.008
   Web of Science ®Google Scholar
• H. Wang, A time-splitting spectral method for coupled Gross-Pitaevskii equations with applications to rotating Bose-Einstein condensates, J. Comput. Appl. Math. 205 (2007), pp. 88–104. doi: 10.1016/j.cam.2006.04.042
   Web of Science ®Google Scholar
• C.J. Wang, C. Gao, C.M. Jian, and H. Zhai, Spin-orbit coupled spinor Bose-Einstein condensates, Phys. Rev. Lett. 105 (2010), p. 160403.
   PubMed Web of Science ®Google Scholar
• T.C. Wang and B.L. Guo, Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimension, Sci. Sin. Math. 41 (2011), pp. 207–233. doi: 10.1360/012010-846
   Google Scholar
• X. Wei, L. Zhang, S. Wang, and F. Liao, An efficient split-step compact finite difference method for the coupled Gross-Pitaevskii equations, Int. J. Comput. Math. 96 (2019), pp. 2334–2351. doi: 10.1080/00207160.2018.1557326
   Google Scholar
• Y. Zhang, L. Mao, and C. Zhang, Mean-field dynamics of spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett. 108 (2012), p. 035302.
   PubMed Web of Science ®Google Scholar
• Q. Zhu, C. Zhang, and B. Wu, Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates, Europhys. Lett. 100 (2012), p. 50003. doi: 10.1209/0295-5075/100/50003
   Google Scholar