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A G-FDTD scheme for solving multi-dimensional open dissipative Gross-Pitaevskii equations. (English) Zbl 1352.65256

Summary: Behaviors of dark soliton propagation, collision, and vortex formation in the context of a non-equilibrium condensate are interesting to study. This can be achieved by solving open dissipative Gross-Pitaevskii equations (dGPEs) in multiple dimensions, which are a generalization of the standard Gross-Pitaevskii equation that includes effects of the condensate gain and loss. In this article, we present a generalized finite-difference time-domain (G-FDTD) scheme, which is explicit, stable, and permits an accurate solution with simple computation, for solving the multi-dimensional dGPE. The scheme is tested by solving a steady state problem in the non-equilibrium condensate. Moreover, it is shown that the stability condition for the scheme offers a more relaxed time step restriction than the popular pseudo-spectral method. The G-FDTD scheme is then employed to simulate the dark soliton propagation, collision, and the formation of vortex-antivortex pairs.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Pitaevskii, L. P.; Stringari, S., Bose-Einstein condensation, (2003), Clarendon Press · Zbl 1110.82002
[2] Succi, S.; Toschi, F.; Tosi, M. P.; Vignolo, P., Bose-Einstein condensates and the numerical solution of the Gross-Pitaevskii equation, Comput. Sci. Eng., 7, 48-57, (2005)
[3] Kasprzak, J.; Richard, M.; Kundermann, S.; Baas, A.; Jeambrun, P.; Keeling, J. M.J.; Szymanska, M. H.; Marchetti, F. M., Bose-Einstein condensation of exciton polaritons, Nature, 443, 409-414, (2006)
[4] Deng, H.; Weihs, G.; Santori, C.; Bloch, J.; Yamamoto, Y., Condensation of semiconductor microcavity exciton polaritons, Science, 298, 199-202, (2002)
[5] Deng, H.; Haug, H.; Yamamoto, Y., Exciton-polariton Bose-Einstein condensation, Rev. Mod. Phys., 82, 1489-1537, (2010)
[6] Amo, A.; Sanvitto, D.; Laussy, F. P.; Ballarini, D.; Del Valle, E.; Martin, M. D.; Lemaitre, A., Collective fluid dynamics of a polariton condensate in a semiconductor microcavity, Nature, 457, 291-295, (2009)
[7] Lagoudakis, K. G.; Wouters, M.; Richard, M.; Baas, A.; Carusotto, I.; André, R.; Dang, Le Si; Deveaud-Plédran, B., Quantized vortices in an exciton-polariton condensate, Nat. Phys., 4, 706-710, (2008)
[8] Roumpos, G.; Fraser, M. D.; Löffler, A.; Höfling, S.; Forchel, A.; Yamamoto, Y., Single vortex-antivortex pair in an exciton-polariton condensate, Nat. Phys., 7, 129-133, (2010)
[9] Lai, C. W.; Kim, N. Y.; Utsunomiya, S.; Roumpos, G.; Deng, H.; Fraser, M. D.; Byrnes, T., Coherent zero-state and pi-state in an exciton-polariton condensate array, Nature, 450, 529-532, (2007)
[10] Masumoto, N.; Kim, N. Y.; Byrnes, T.; Kusudo, K.; Löffler, A.; Höfling, S.; Forchel, A.; Yamamoto, Y., Exciton-polariton condensates with flat bands in a two-dimensional Kagome lattice, New J. Phys., 14, 065002, (2012)
[11] Wouters, M.; Carusotto, I., Excitations in a nonequilibrium Bose-Einstein condensate of exciton polaritons, Phys. Rev. Lett., 99, 140402, (2007)
[12] Keeling, J.; Berloff, N. G., Spontaneous rotating vortex lattices in a pumped decaying condensate, Phys. Rev. Lett., 100, 250401, (2008)
[13] Wouters, M.; Carusotto, I., Superfluidity and critical velocities in nonequilibrium Bose-Einstein condensates, Phys. Rev. Lett., 105, 020602, (2010)
[14] Byrnes, T.; Horikiri, T.; Ishida, N.; Fraser, M.; Yamamoto, Y., Negative Bogoliubov dispersion in exciton-polariton condensates, Phys. Rev. B, 85, 075130, (2012)
[15] Minguzzi, A.; Succi, S.; Toschi, F.; Tosi, M. O.; Vignolo, P., Numerical methods for atomic quantum gases with applications to Bose-Einstein condensates and to ultracold fermions, Phys. Rep., 395, 223-355, (2004)
[16] Muruganandam, P.; Adhikari, S. K., Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 180, 1888-1912, (2009) · Zbl 1353.35002
[17] Choi, Y. S.; Javanainen, J.; Koltracht, I., A fast algorithm for the solution of the time-independent Gross-Pitaevskii equation, J. Comput. Phys., 190, 1-21, (2003) · Zbl 1027.65157
[18] Adhikari, S. K., Numerical study of the spherically symmetric Gross-Pitaevskii equation in two space dimensions, Phys. Rev. E, 62, 2937-2944, (2000)
[19] Adhikari, S. K., Numerical study of the coupled time-dependent Gross-Pitaevskii equation: application to Bose-Einstein condensation, Phys. Rev. E, 63, 056704, (2001)
[20] Adhikari, S. K.; Muruganandam, P., Bose-Einstein condensation dynamics from the numerical solution of the Gross-Pitaevskii equation, J. Phys. B, 35, 2831-2843, (2002)
[21] Holland, M. J.; Cooper, J., Expansion of a Bose-Einstein condensate in a harmonic potential, Phys. Rev. A, 53, R1954-R1957, (1996)
[22] Vudragovic, D.; Vidanovic, I.; Balaz, A.; Muruganandam, P.; Adhikari, S. K., C programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap, Comput. Phys. Commun., 183, 2021-2025, (2012) · Zbl 1353.35003
[23] Bao, W.; Cai, Y., Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comput., 82, 99-128, (2013) · Zbl 1264.65146
[24] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009
[25] Wang, T. C.; Zhao, X. F., Optimal \(l^\infty\) error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions, Sci. China Math., 57, (2014) · Zbl 1320.65130
[26] Cerimele, M. M.; Chiofalo, M. L.; Pistella, F.; Succi, S.; Tosi, M. P., Numerical solution of the Gross-Pitaevskii equation using an explicit finite-difference scheme: an application to trapped Bose-Einstein condensates, Phys. Rev. E, 62, 1382-1388, (2000)
[27] Edwards, M.; Dodd, R. J.; Clark, C. W.; Ruprecht, P. A.; Burnett, K., Properties of a Bose-Einstein condensate in an anisotropic harmonic potential, Phys. Rev. A, 53, R1950, (1996)
[28] Dalfovo, F.; Stringari, S., Bosons in anisotropic traps: ground state and vortices, Phys. Rev. A, 53, 247, (1996)
[29] Chiofalo, M. L.; Succi, S.; Tosi, M. P., Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev. E, 62, 7438-7443, (2000)
[30] Moxley, F. I.; Byrnes, T.; Fujiwara, F.; Dai, W., A generalized finite-difference time-domain quantum method for the N-body interacting Hamiltonian, Comput. Phys. Commun., 183, 2434-2440, (2012) · Zbl 1302.65197
[31] Moxley, F. I.; Chuss, D. T.; Dai, W., A generalized finite-difference time-domain scheme for solving nonlinear Schrödinger equations, Comput. Phys. Commun., 184, 1834-1841, (2013) · Zbl 1344.65083
[32] Visscher, P. B., A fast explicit algorithm for the time-dependent Schrödinger equation, Comput. Phys., 5, 596, (1991)
[33] Morton, K. W.; Mayers, D. F., Numerical solution of partial differential equations, (1994), Cambridge University Press · Zbl 1126.65077
[34] Yang, J., Nonlinear waves in integrable and non-integrable systems, (2010), SIAM
[35] Kivshar, Y. S.; Luther-Davies, B., Dark optical solitons: physics and applications, Phys. Rep., 298, 81-197, (1998)
[36] Muryshev, A. E.; van den Heuvell, H. V.L.; Shlyapnikov, G. V., Stability of standing matter waves in a trap, Phys. Rev. A, 60, R2665, (1999)
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