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

Öz

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.

Anahtar Kelimeler

• Coupled system,trapping potential,finite difference method

Tuzaklama potansiyelli iki-boyutlu zaman-yer kesirli türevli Schrödinger denklemlerinin bağlı sistemlerinin sayısal analizi

Öz

Bu çalışmada zaman ve yer boyutlarında kesirli türevli Schrödinger diferansiyel denklemlerinin bağıl sistemlerinin genel ve klasik formları tuzaklama potansiyeli altında sayısal bir yaklaşımla ele alınmıştır. Bu tip problemlerin fark şemalarının kararlılıkları üzerine teoremler kurulmuş ve ispatlarıyla sunulmuştur. Sayısal sonuçlar tek ve iki boyutlu durumlar için incelenmiştir. Yaklaşım mertebeleri sayısal deneylerle ispatlanmıştır. Çalışma boyunca tuzaklama potansiyelinin bu tip sistemler üzerine etkisi araştırılmıştır.

Anahtar Kelimeler

• Bağlı sistem,tuzaklama potansiyeli,sonlu farklar metodu

Kaynakça

1. 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).
2. 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).
3. 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).
4. Hicdurmaz B. and Ashyralyev, A., On the stability of time-fractional Schrödinger diferential equations, Numerical Functional Analysis and Optimization, 38, 1215-1225, (2017).
5. 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).
6. 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).
7. Antoine, X., Besse, C, and Klein, P., Absorbing boundary conditions for general nonlinear Schrödinger equations, SIAM Journal on Scientific Computing, 33, 1008-1033, (2011).
8. 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).

9. 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).
10. 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).
11. 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).
12. 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).
13. 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).
14. 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)
15. 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).
16. Kirkpatrick, K., Zhang, Y., Fractional Schrödinger dynamics and decoherence, Physica D: Nonlinear Phenomena, 332, 41-54, (2016).
17. 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).
18. 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).
19. Ashyralyev A., A note on fractional derivatives and fractional powers of operators, Journal of Mathematical Analysis and Applications, 357, 232-236, (2009).
20. 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).