Numerical analysis for coupled systems of two-dimensional time-space fractional Schrödinger equations with trapping potentials

Betül Hiçdurmaz


In this study general and classical coupled systems of nonlinear time-space fractional Schrödinger equations (TSFSDE) with trapping potentials are investigated with a numerical approach. Theorems on stability of the finite difference schemes for such problems are established and presented with their proofs. Numerical solutions are investigated for one and two-dimensional cases. Convergence rates are proved by numerical experiments. Effect of a trapping potential on such systems is searched throughout the paper.

Tam Metin:



Lü, X. and Peng, M., Painlevé-integrability and explicit solutions of the general two-coupled nonlinear Schrödinger system in the optical fiber communications, Nonlinear Dynamics, 73, 405-410, (2013).

Wang, D. S., Zhang, D. J. and Yang, J., Integrable properties of the general coupled nonlinear Schrödinger equations, Journal of Mathematical Physics, 51, Article ID 023510, (2010).

Ding, H.F., Li, C.P. and Chen, Y.Q., High-order algorithms for Riesz derivative and their applications, Abstract and Applied Analysis, 653797, 19- 55, (2014).

Hicdurmaz B. and Ashyralyev, A., On the stability of time-fractional Schrödinger differential equations, Numerical Functional Analysis and Optimization, 38, 1215-1225, (2017).

Wang, D., Xiao, A. and Yang, W. , A linearly implicit con-servative difference scheme for the space fractional coupled nonlinear Schrödinger equations, Journal of Computational Physics, 272, 644-655 (2014).

Garrappa, R., Moret, I. and Popolizio, M., On the time-fractional Schrödinger equation: theoretical analysis and numerical solution by matrix Mittag-Leffler functions, Computers and Mathematics with Applications, 74, 977-992 (2017).

Antoine, X., Besse, C, and Klein, P., Absorbing boundary conditions for general nonlinear Schrödinger equations, SIAM Journal on Scientific Computing, 33, 1008-1033, (2011).

Yuan, Y. Q., Tian, B., Liu, L. and Sun, Y., Bright-dark solitons for a set of the general coupled nonlinear Schrödinger equations in a birefringent fiber, Europhysics Letters, 120, 30001, 1-5, (2017).

Ashyralyev A. and Sirma, A., A note on the numerical solu-tion of the semilinear Schrödinger equation, Nonlinear Analysis: Theory, Methods and Applications, 71, 12, 2507-2516, (2009).

Al-Hashimi, N.H.N. and Ghalib, S. K., Theoretical analysis of different external trapping potential used in experimental of BEC, IOSR Journal of Engineering, 2, 11, 1-5, (2012).

Hamed, S. H. M., Yousif, E. A. and Arbab, A. I., Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation Sumudu transform method, Abstract and Applied Analysis, 2014, 863015, 1-13, (2014).

Khan, N. A., Jamil M., and Ara, A., Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method, ISRN Mathematical Physics, 2012, 197068, 1-11, (2012).

Taghizadeh, N., Noori S. R. M., Exact solutions of the cubic nonlinear Schrödinger equation with a trapping potential by reduced differential transform method, Mathematical Scientific Letters, 5, 3, 297-302, (2016).

Hicdurmaz, B., Multidimensional problems for general coupled systems of time-space fractional Schrödinger equations, Journal of Coupled Systems and Multiscale Dynamics, 6, 147-153 (2018)

Antoine, X., Tang, Q., and Zhang, Y., On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions, Journal of Computational Physics, 325, 74-97, (2016).

Kirkpatrick, K., Zhang, Y., Fractional Schrödinger dynamics and decoherence, Physica D: Nonlinear Phenomena, 332, 41-54, (2016).

Wang, M., Shan, W. R., Lü, X., Xue, Y.S., Lin, Z.Q. and Tian B., Soliton collision in a general coupled nonlinear Schrödinger system via symbolic computation, Applied Mathematics and Computation 219, 11258-11264, (2013).

Liu, Q., Zeng, F. and Li, C., Finite difference method for time-space-fractional Schrödinger equation, International Journal of Computer Mathematics, 92, 7, 1439-1451, (2015).

Ashyralyev A., A note on fractional derivatives and fractional powers of operators, Journal of Mathematical Analysis and Applications, 357, 232- 236, (2009).

Tian, W.Y., Zhou, H. and Deng W.H., A class of second order difference approximation for solving space fractional diffusion equations, Mathematics of Computation,84,1703-1727, (2015).


  • Şu halde refbacks yoktur.

Telif Hakkı (c) 2020 Betül Hiçdurmaz

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.