×

zbMATH — the first resource for mathematics

Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. (English) Zbl 1082.65570
Summary: We use the split-step finite difference method to solve various nonlinear Schrödinger equations including coupled ones. We study the numerical accuracy of the method. Detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear Schrödinger equations.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Muruganandam, P.; Adhikari, S.K, Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods, J. phys. B, 36, 2501-2513, (2003)
[2] Bao, W.; Jaksch, D.; Markowich, P.A., Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. comput. phys., 187, 1, 318-342, (2003) · Zbl 1028.82501
[3] W. Bao, H. Wang, P.A. Markowich, Ground, symmetric and central vortex states in rotating Bose-Einstein condensates, Preprint. · Zbl 1073.82004
[4] Bandrauk, A.D.; Shen, H., Exponential split operator methods for solving coupled time-dependent Schrödinger equations, J. chem. phys., 99, 2, 1185-1193, (1993)
[5] Bandrauk, A.D.; Shen, H., High-order split-step exponential methods for solving coupled nonlinear Schrödinger equations, J. phys. A-math. gen., 27, 7147-7155, (1994) · Zbl 0842.65084
[6] 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, 1382-1389, (2000)
[7] Chang, Q.; Jia, E.; Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. comput. phys., 148, 397-415, (1999) · Zbl 0923.65059
[8] Dion, C.M.; Cances, E., Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap, Phys. rev. E, 67, 046706-1-046706-9, (2003)
[9] Deconinck, B.; Frigyik, B.A.; Kutz, J.N., Stability of exact solutions of the defocusing nonlinear Schrödinger equation with periodic potential in two dimensions, Phys. lett. A, 283, 177-184, (2001) · Zbl 0984.81028
[10] Deconinck, B.; Frigyik, B.A.; Kutz, J.N., Dynamics and stability of Bose-Einstein condensates: the nonlinear Schrödinger equation with periodic potential, J. nonlinear sci., 12, 169-205, (2002) · Zbl 1009.35078
[11] Fairweather, G.; Khebchareon, M., Numerical methods for schrodinger-type problems, (), 219-250
[12] Gardner, L.R.T.; Gardner, G.A.; Zaki, S.I.; El Sahrawi, Z., B-spline finite element studies of the non-linear schriidinger equation, Comput. methods appl. mech. eng., 108, 303-318, (1993) · Zbl 0842.65083
[13] Ismail, M.S.; Thiab, R., Taha numerical simulation of coupled nonlinear Schrödinger equation, Math. comput. simul., 56, 547-562, (2001) · Zbl 0972.78022
[14] Kasamatsu, K.; Tsubota, M.; Ueda, M., Structure of vortex lattices in rotating two-component Bose-Einstein condensates, Physica B, 329-333, 23-24, (2003)
[15] Morton, K.W.; Mayers, D.F., Numerical solution of partial differential equations, (1994), Cambridge University Press Cambridge · Zbl 0811.65063
[16] M.C. Lai, C.Y. Huang, a simple Dufort-Frankel type scheme for the Gross-Pitaevskii equation of Bose-Einstein condensates on different geometries, preprint. · Zbl 1050.81079
[17] Perez-Garcia, V.M.; Liu, X.Y., Numerical methods for the simulation of trapped nonlinear Schrödinger systems, Appl. math. comput., 144, 2-3, 215-235, (2003) · Zbl 1024.65084
[18] Sulem, P.L.; Sulem, C.; Patera, A., Numerical simulation of singular solutions to the two-dimensional cubic Schrödinger equation, Commun. pure appl. math., 37, 755-778, (1984) · Zbl 0543.65081
[19] Sulem, C.; Sulem, P.L., The nonlinear Schrödinger equation, self-focusing and wave collapse, (1999), Springer New York · Zbl 0928.35157
[20] Taha, T.R., A numerical scheme for the nonlinear Schrödinger equation, Comput. math. appl., 22, 9, 77-84, (1991) · Zbl 0755.65130
[21] Weideman, J.A.C.; Herbst, B.M., Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. numer. anal., 23, 3, 485-507, (1986) · Zbl 0597.76012
[22] Yoshida, H., Construction of higher order symplectic integrators, Phys. lett. A, 150, 12, 262-268, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.