References
- X. Antoine, Q.L. Tang, and Y. Zhang, On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys. 325 (2016), pp. 74–97. https://doi.org/10.1016/j.jcp.2016.08.009.
- W.Z. Bao and Y.Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic Related Models 6(1) (2013), pp. 1–135. https://doi.org/10.3934/krm.2013.6.1.
- W.Z. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J on Scientific Comput 25(5) (2004), pp. 1674–1697. https://doi.org/10.1137/S1064827503422956.
- A.H. Bhrawy and M.A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, J. Comput. Phys. 294 (2015), pp. 462–483. https://doi.org/10.1016/j.jcp.2015.03.063.
- G.M. Bisci, V.D. Radulescu, and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge, Cambridge University Press, 2016.
- H.J. Bungartz and M. Griebel, Sparse grids, Acta Numer. 13 (2004), pp. 147–269. https://doi.org/10.1017/S0962492904000182.
- W.T. Cai, D.D. He, and K.J. Pan, A linearized energy-conservative finite element method for the nonlinear Schrödinger equation with wave operator, Appl. Numer. Math. 140 (2019), pp. 183–198. https://doi.org/10.1016/j.apnum.2019.02.005.
- X.J. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Phys. 54 (2013), pp. 061504. https://doi.org/10.1063/1.4809933.
- J.P. Dong, Applications of density matrix in the fractional quantum mechanics: Thomas–Fermi model and Hohenberg–Kohn theorems revisited, Phys. Lett. A 375(30–31) (2011), pp. 2787–2792. https://doi.org/10.1016/j.physleta.2011.05.055.
- S.W. Duo and Y.Z. Zhang, Applications of density matrix in the fractional quantum mechanics: Thomas–Fermi model and Hohenberg–Kohn theorems revisited Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl. 71(11) (2016), pp. 2257–2271. https://doi.org/10.1016/j.camwa.2015.12.042.
- M.F. Fei, C.M. Huang, and P.D. Wang, Error estimates of structure-preserving Fourier pseudospectral methods for the fractional Schrödinger equation, Numer. Meth. Partial. Differ. Equ 36(2) (2020), pp. 369–393. https://doi.org/10.1002/num.22432.
- V. Gradinaru, Fourier transform on sparse grids: code design and the time dependent Schrödinger equation, Computing 80(1) (2007), pp. 1–22. https://doi.org/10.1007/s00607-007-0225-3.
- M. Griebel and J. Hamaekers, Sparse grids for the Schrödinger equation, ESAIM: Math. Model. Numer. Anal. 41(2) (2007), pp. 215–247. https://doi.org/10.1051/m2an:2007015.
- M. Griebel and J Hamaekers, Fast discrete Fourier transform on generalized sparse grids. In: J. Garcke & D. Pflüger, eds., Sparse Grids and Applications-Munich 2012, Springer International Publishing, 2014, pp. 75–107.
- A. Guerrero and M.A. Moreles, On the numerical solution of the eigenvalue problem in fractional quantum mechanics, Commun. Nonlinear Sci. Numer. Simul. 20(2) (2015), pp. 604–613. https://doi.org/10.1016/j.cnsns.2014.06.013.
- B.L. Guo, Y.Q. Han, and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput. 204(1) (2008), pp. 468–477. https://doi.org/10.1016/j.amc.2008.07.003.
- K. Hallatschek, Fouriertransformation auf dünnen gittern mit hierarchischen basen, Numer. Math.63(1) (1992), pp. 83–97. https://doi.org/10.1007/BF01385849.
- Y.Q. Huang, X.Y. Li, and A.G. Xiao, Fourier pseudospectral method on generalized sparse grids for the space-fractional Schrödinger equation, Comput. Math. Appl. 75(12) (2018), pp. 4241–4255. https://doi.org/10.1016/j.camwa.2018.03.026.
- C. Klein, C. Sparber, and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. Royal Soc. A Math. Phys. Eng. Sci. 470 (2014), pp. 20140364. https://doi.org/10.1098/rspa.2014.0364.
- N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62(3) (2000), pp. 3135–3145. https://doi.org/10.1103/PhysRevE.62.3135.
- M. Li, C.M. Huang, and W.Y. Ming, A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations, Numer. Algorith. 83(1) (2020), pp. 99–124. https://doi.org/10.1007/s11075-019-00672-3.
- M. Li, C.M. Huang, and P.D. Wang, Galerkin finite element method for nonlinear fractional Schrödinger equations, Numer. Algorith. 74(2) (2017), pp. 499–525. https://doi.org/10.1007/s11075-016-0160-5.
- Z.P. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains, SIAM J. Sci. Comput. 39(5) (2017), pp. A1928–A1950. https://doi.org/10.1137/16M1097109.
- J. Shen and H.J. Yu, Efficient spectral sparse grid methods and applications to high-dimensional elliptic equations II, unbounded domains, SIAM J. Sci. Comput. 34(2) (2012), pp. A1141–A1164. https://doi.org/10.1137/110834950.
- P.D. Wang and C.M Huang, Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math. 129 (2018), pp. 137–158. https://doi.org/10.1016/j.apnum.2018.03.008.
- T.C. Wang, B.L. Guo, and L.M. Zhang, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput. 217(4) (2010), pp. 1604–1619. https://doi.org/10.1016/j.amc.2009.07.040.
- D.L. Wang, A.G. Xiao, and W. Yang, Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative, J. Comput. Phys. 242 (2013), pp. 670–681. https://doi.org/10.1016/j.jcp.2013.02.037.
- D.L. Wang, A.G. Xiao, and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys. 272 (2014), pp. 644–655. https://doi.org/10.1016/j.jcp.2014.04.047.
- D.L. Wang, A.G. Xiao, and W. Yang, Maximum-norm error analysis of a difference scheme for the space fractional CNLS, Appl. Math. Comput. 257 (2015), pp. 241–251. https://doi.org/10.1016/j.amc.2014.11.026.