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On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions. (English) Zbl 1380.65296
Summary: In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of fractional Schrödinger equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation [L. Exl et al., ibid. 327, 629–642 (2016; Zbl 1422.65450)]. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method [the first author and R. Duboscq, ibid. 258, 509–523 (2014; Zbl 1349.82027)] and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform [W. Bao et al., SIAM J. Sci. Comput. 35, No. 6, A2671–A2695 (2013; Zbl 1286.35213)], we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE. In parallel to the numerical schemes, we also prove some existence and nonexistence results for the ground states. Dynamical laws of some standard quantities, including the mass, energy, angular momentum and the center of mass, are stated. The ground states properties with respect to the fractional order and/or rotating frequencies, dynamics involving decoherence and turbulence together with some interesting phenomena are reported.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Software:
GPELab
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[1] Aki, G. L.; Markowich, P. A.; Sparber, C., Classical limit for semi-relativistic Hartree system, J. Math. Phys., 49, 102-110, (2008)
[2] Antoine, X.; Arnold, A.; Besse, C.; Ehrhardt, M.; Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4, 729-796, (2008) · Zbl 1364.65178
[3] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184, 2621-2633, (2013) · Zbl 1344.35130
[4] Antoine, X.; Duboscq, R., Robust and efficient preconditioned Krylov spectral solvers for computing the ground states of fast rotating and strongly interacting Bose-Einstein condensates, J. Comput. Phys., 258, 509-523, (2014) · Zbl 1349.82027
[5] Antoine, X.; Duboscq, R., Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations I: computation of stationary solutions, Comput. Phys. Commun., 185, 2969-2991, (2014) · Zbl 1348.35003
[6] Antoine, X.; Duboscq, R., Gpelab, a Matlab toolbox to solve Gross-Pitaevskii equations II: dynamics and stochastic simulations, Comput. Phys. Commun., 193, 95-117, (2015) · Zbl 1344.82004
[7] Antoine, X.; Duboscq, R., Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity, (Nonlinear Optical and Atomic Systems: at the Interface of Mathematics and Physics, Lecture Notes in Mathematics, vol. 2146, (2015), Springer), 49-145 · Zbl 1344.35114
[8] Antonelli, P.; Marahrens, D.; Sparber, C., On the Cauchy problem for nonlinear Schrödinger equations with rotation, Discrete Contin. Dyn. Syst., Ser. A, 32, 703-715, (2012) · Zbl 1234.35238
[9] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135, (2013) · Zbl 1266.82009
[10] Bao, W.; Cai, Y.; Wang, H., Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229, 7874-7892, (2010) · Zbl 1198.82036
[11] Bao, W.; Dong, X., Numerical methods for computing ground state and dynamics of nonlinear relativistic Hartree equation for boson stars, J. Comput. Phys., 230, 5449-5469, (2011) · Zbl 1220.83023
[12] Bao, W.; Jian, H.; Mauser, N. J.; Zhang, Y., Dimension reduction of the Schrödinger equation with Coulomb and anisotropic confining potentials, SIAM J. Appl. Math., 73, 2100-2123, (2013) · Zbl 1294.35130
[13] Bao, W.; Jiang, S.; Tang, Q.; Zhang, Y., Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296, 72-89, (2015) · Zbl 1354.65200
[14] Bao, W.; Marahrens, D.; Tang, Q.; Zhang, Y., A simple and efficient numerical method for computing the dynamics of rotating Bose-Einstein condensates via rotating Lagrangian coordinates, SIAM J. Sci. Comput., 35, A2671-A2695, (2013) · Zbl 1286.35213
[15] Bao, W.; Tang, Q.; Xu, Z., Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys., 235, 423-445, (2013) · Zbl 1291.35329
[16] Bao, W.; Tang, Q.; Zhang, Y., Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT · Zbl 1388.65115
[17] Bayin, S. S., On the consistency of solutions of the space fractional Schrödinger equation, J. Math. Phys., 53, (2012) · Zbl 1275.81026
[18] Bayin, S. S., Comment on “on the consistency of solutions of the space fractional Schrödinger equation”, J. Math. Phys., 54, (2013) · Zbl 1284.81104
[19] C. Besse, G. Dujardin, I. Lacroix-Violet, High-order exponential integrators for nonlinear Schrödinger equations with application to rotating Bose-Einstein condensates, 2015, \(\langle \text{hal-01170888} \rangle\). · Zbl 1371.35235
[20] Carles, R.; Markowich, P. A.; Sparber, C., On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21, 2569-2590, (2008) · Zbl 1157.35102
[21] Cho, Y.; Hajaiej, H.; Hwang, G.; Ozawa, T., On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkc. Ekvacioj, 56, 193-224, (2013) · Zbl 1341.35138
[22] Cho, Y.; Hajaiej, H.; Hwang, G.; Ozawa, T., On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13, 1267-1282, (2014) · Zbl 1282.35319
[23] Cho, Y.; Ozawa, T., On the semi-relativistic Hartree-type equation, SIAM J. Math. Anal., 38, 1060-1074, (2006) · Zbl 1122.35119
[24] Coti Zelati, V.; Nolasco, M., Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations, Rend. Lincei, Mat. Appl., 22, 51-72, (2011) · Zbl 1219.35292
[25] Danaila, I.; Kazemi, P., A new Sobolev gradient method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32, 2447-2467, (2010) · Zbl 1216.35006
[26] Du, Q.; Gunzburger, M.; Lehoucq, R. B.; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54, 667-696, (2012) · Zbl 1422.76168
[27] Duo, S.; Zhang, Y., Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl., 71, 11, 2257-2271, (2016)
[28] Elgart, A.; Schlein, B., Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60, 500-545, (2007) · Zbl 1113.81032
[29] Ertik, H.; Demirhan, D.; Sirin, H.; Buyukkilic, F., Time fractional development of quantum systems, J. Math. Phys., 51, (2010) · Zbl 1312.81080
[30] Ertik, H.; Sirin, H.; Demirhan, D.; Buyukkilic, F., Fractional mathematical investigation of Bose-Einstein condensation in dilute ^87rb, ^23na and ^7Li atomic gases, Int. J. Mod. Phys. B, 26, 1250096, (2012) · Zbl 1257.82080
[31] Exl, L.; Mauser, N. J.; Zhang, Y., Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation · Zbl 1422.65450
[32] Feynman, R. P.; Hibbs, A. R., Quantum mechanics and path integrals, (1965), McGraw-Hill New York · Zbl 0176.54902
[33] Frank, R. L.; Lenzmann, E., Uniqueness of nonlinear ground states for fractional Laplacians in \(\mathbb{R}\), Acta Math., 210, 261-318, (2013) · Zbl 1307.35315
[34] Fröhlich, J.; Lenzmann, E., Blowup for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math., 60, 1691-1705, (2007) · Zbl 1135.35011
[35] Guo, B.; Pu, X.; Huang, F., Fractional partial differential equations and their numerical solutions, (2015), World Scientific Singapore
[36] Hawkins, E.; Schwarz, J. M., Comment on “on the consistency of solutions of the space fractional Schrödinger equation”, J. Math. Phys., 54, (2013) · Zbl 1280.81040
[37] Henry, B. I.; Langlands, T. A.M.; Straka, P., An introduction to fractional diffusion, (Dewar, R. L.; Detering, F., Complex Physical, Biophysical and Econophysical Systems, World Scientific Lecture Notes in Complex Systems, vol. 9, (2010), World Scientific Hackensack, NJ) · Zbl 1221.60047
[38] Hong, Y.; Sire, Y., On fractional Schrödinger equation in Sobolev spaces, Commun. Pure Appl. Anal., 14, 2265-2282, (2015) · Zbl 1338.35466
[39] Ionescu, A. D.; Pusateri, F., Nonlinear fractional Schrödinger equations in one dimension, J. Funct. Anal., 266, 139-176, (2014) · Zbl 1304.35749
[40] Jeng, M.; Xu, S.-L.-Y.; Hawkins, E.; Schwarz, J. M., On the nonlocality of the fractional Schrödinger equation, J. Math. Phys., 51, (2010) · Zbl 1311.81114
[41] Jiang, S.; Greengard, L.; Bao, W., Fast and accurate evaluation of dipolar interaction in Bose-Einstein condensates, SIAM J. Sci. Comput., 36, B777-B794, (2014) · Zbl 1307.65184
[42] Karniadakis, G. E.; Hesthaven, J. S.; Podlubny, I., Fractional PDEs theory, numerics and applications, J. Comput. Phys., 293, 1-462, (2015)
[43] Kirkpatrick, K.; Lenzmann, E.; Staffilan, G., On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317, 563-591, (2012)
[44] K. Kirkpatrick, Y. Zhang, Fractional Schrödinger dynamics and decoherence, 2014, preprint.
[45] Klein, C.; Sparber, C.; Markowich, P., Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. A, 470, 20140364, (2014) · Zbl 1372.65284
[46] Laskin, N., Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298-304, (2000) · Zbl 0948.81595
[47] Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, (2002)
[48] Laskin, N., Principles of fractional quantum mechanics · Zbl 1297.81070
[49] Lieb, E. H.; Loss, M., Analysis, (2001), American Mathematical Society · Zbl 0966.26002
[50] Naber, M., Time fractional Schrödinger equation, J. Math. Phys., 45, 3339, (2004) · Zbl 1071.81035
[51] Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering, vol. 198, (1999), Academic Press · Zbl 0918.34010
[52] Secchi, S., Ground state solutions for nonlinear fractional Schrödinger equation in \(\mathbb{R}^N\), J. Math. Phys., 54, (2013) · Zbl 1281.81034
[53] Secchi, S.; Squassina, M., Soliton dynamics for fractional Schrödinger equation, Appl. Anal., 93, 1702-1729, (2014) · Zbl 1298.35166
[54] Shang, X.; Zhang, J., Ground state for fractional Schrödinger equation with critical growth, Nonlinearity, 27, 187-207, (2014) · Zbl 1287.35027
[55] Tang, Q., Numerical studies on quantized vortex dynamics in superfluidity and superconductivity, (2013), National University of Singapore, Ph. D thesis
[56] Uzar, N.; Ballikaya, S., Investigation of classical and fractional Bose-Einstein condensation for harmonic potential, Physica A, 392, 1733-1741, (2013)
[57] Uzar, N.; Han, S. D.; Tufekci, T.; Aydiner, E., Solutions of the Gross-Pitaevskii and time fractional Gross-Pitaevskii equations for different potentials with homotopy perturbation method
[58] Wang, P.; Huang, C., An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys., 293, 238-251, (2015) · Zbl 1349.65346
[59] Wang, S.; Xu, M., Generalized fractional Schrödinger equation with space-time fractional derivatives, J. Math. Phys., 48, (2007) · Zbl 1137.81328
[60] Zhang, Y.; Dong, X., On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230, 2660-2676, (2011) · Zbl 1218.65115
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