References
- Bao W, Cai Y. Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J Numer Anal. 2012;50:492–521.
- Machihara S, Nakanishi K, Ozawa T. Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations. Math Ann. 2002;322:603–621.
- Schoene A. On the nonrelativistic limits of the Klein-Gordon and Dirac equations. J Math Anal Appl. 1979;71:36–47.
- Tsutsumi M. Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions. Nonlinear Anal-Theor. 1984;8:637–643.
- Berg L, Colin T. A singular perturbation problem for an envelope equation in plasma physics. Physica D. 1995;84:437–459.
- Colin T, Fabrie P. Semidiscretization in time for nonlinear Schrödinger-waves equations. Discret Contin Dyn Syst. 1998;4:671–690.
- Bao W, Dong X, Xin J. Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse. Physica D. 2010;239:1120–1134.
- Xin J. Modeling light bullets with the two-dimensional sine-Gordon equation. Physica D. 2000;135:345–368.
- Laskin N. Fractional quantum mechanics. Phys Rev E. 2000;62:3135–3145.
- Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions. Commun Math Phys. 2013;317:563–591.
- Fröhlich J, Jonsson BLG, Lenzmann E. Boson stars as solitary waves. Commun Math Phys. 2007;274:1–30.
- Lenzmann E. Well-posedness for semi-relativistic Hartree equations of critical type. Math Phys Anal Geom. 2007;10:43–64
- Ionescu AD, Pusateri F. Nonlinear fractional Schrödinger equations in one dimension. J Funct Anal. 2014;266:139–176.
- Numerical simulation of nonlinear Schrödinger equation system: a new conservative scheme. Appl Math Comput. 1995;71:165-–177.
- He D, Pan K. An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions. Comput Math Appl. 2017;73:2360–2374.
- Thalhammer M. High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J Numer Anal. 2008;26:2022–2038.
- Bao W, Jaksch D, Markowich P. Numercial solution of the Gross-Pitaevskii equation for Bose-Einstein condensation. J Comput Phys. 2003;187:318–342.
- Bao W, Li H, Shen J. A generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating bose-Einstein condensates. SIAM J Sci Comput. 2009;31:3685–3711.
- Hu H, Xie S. A high accurate and conservative difference scheme for a class of nonlinear Schrödinger equation with wave operator. Appl Math-J Chin Univ Ser B. 2014;29:36–43.
- Zhang L, Chang Q. A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator. Appl Math Comput. 2003;145:603–612.
- Wang T, Zhang L. Analysis of some new conservative scheme for nonlinear Schrödinger equation with wave operator. Appl Math Comput. 2006;182:1780–149.
- Li X, Zhang L, Wang S. A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Appl Math Comput. 2012;219:3187–3197.
- Wang P, Huang C. An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J Comput Phys. 2015;293:238–251.
- Wang P, Huang C. A conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Numer Algorithms. 2015;69:625–641.
- Wang P, Huang C, Zhao L. Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation. J Comput Appl Math. 2016;306:231–247.
- Pan K, Jin X, He D. Pointwise error estimates of a linearized difference scheme for strongly coupled fractional Ginzburg-Landau equations. Math Methods Appl Sci. 2020;43:512–535.
- Zhao X, Sun Z, Hao Z. A fourth-order compact ADI scheme for two-Dimensional nonlinear space fractional schrödinger equation. SIAM J Sci Comput. 2014;36:A2865–A2886.
- Wang D, Xiao A, Yang W. Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J Comput Phys. 2013;242:670–681.
- Ran M, Zhang C. A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator. Int J Comput Math. 2015;93:1103–1118.
- Li M, Zhao Y. A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator. Appl Math Comput. 2018;338:758–773.
- Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions. Commun Math Phys. 2013;317:563–591.
- Wang P, Huang C. An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation. J Comput Phys. 2016;312:31–49.
- Sun H, Sun Z, Gao G. Some high order difference schemes for the space and time fractional Bloch-Torrey equations. Appl Math Comput. 2016;281:356–380.
- Tian W, Zhou H, Deng W. A class of second order difference approximation for solving space fractional diffusion equations. Math Comput. 2015;84:1703–1727.
- Zhou H, Tian W, Deng W. Quasi-compact finite difference schemes for space fractional diffusion equations. J Sci Comput. 2013;56:45–66.
- Yang Q, Liu F, Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl Math Model. 2010;34:200–218.
- Ortigueira M. Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci. 2006;48391:1–12.
- Celik C, Duman M. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J Comput Phys. 2012;231:1743–1750.
- Yue X, Shu S, Xu X, et al. Parallel-in-time multigrid for space–time finite element approximations of two-dimensional space-fractional diffusion equations. Comput Math Appl. 2019;78:3471–3484.
- He D, Pan K. An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau. Numer Algorithms. 2018;79:899–925.
- He D, Pan K. Maximum norm error analysis of an unconditionally stable semi-implicit scheme for multi-dimensional Allen-Cahn equations. Numer Methods Partial Differ Equ. 2019;35:955–975.
- Cai W, He D, Pan K. A linearized energy-conservative finite element method for the nonlinear Schrödinger equation with wave operator. Appl Numer Math. 2019;140:183–198.
- He D, Pan K, Yue X. A positivity preserving and free energy dissipative difference scheme for the poisson-Nernst-Planck system. J Sci Comput. 2019;81:436–458.
- Holte JM. Discrete Gronwall lemma and applications. MAA-NCS Meeting at the University of North Dakota. 2009;24:1–7.
- Zhang X, Sun Z, Wang T. Convergence analysis of a linearied Crank-Nicolson scheme for the two-dimensional complex Ginzburg-Landau equation. Numer Methods Partial Differ Equ. 2013;29:1487–1503.
- Yin B, Wang J, Liu Y, et al. A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schrödinger equations. J Comput Phys. 2020;109869.
- Zhang Q, Lin X, Pan K, et al. Linearized ADI schemes for two-dimensional space-fractional nonlinear Ginzburg-Landau equation. Comput Math Appl. 2020;80:1201–1220.