A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations

Journal of Computational Physics

Volumn 272, Issue , 2014, Pages 644-655

  Wang, Dongling ^a,b   Xiao, Aiguo ^b   Yang, Wei ^b  

^a NORTHWEST UNIVERSITY   (China)

^b XIANGTAN UNIVERSITY   (China)

Author keywords

Fractional centered difference; Fractional Schr dinger equations; Linearly implicit conservative scheme

Indexed keywords

NONLINEAR EQUATIONS;

CONSERVATIVE DIFFERENCE SCHEMES; CONSERVATIVE SCHEMES; DIFFERENCE SOLUTIONS; DISCRETE LEVELS; ENERGY; EXISTENCE AND UNIQUENESS; FRACTIONAL CENTERED DIFFERENCE; FRACTIONAL DERIVATIVES; FRACTIONAL SCHRÖDINGE EQUATION; LINEARLY IMPLICIT CONSERVATIVE SCHEME;

LINEAR SYSTEMS;

EID: 84900805789     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.04.047     Document Type: Article

References (36)

1. Laskin N. Fractional quantum mechanics. Phys. Rev. E 2000, 62:3135-3145.
2. Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 2000, 268(4-6):298-305.
3. Laskin N. Fractional Schrödinger equation. Phys. Rev. E 2002, 66. 05618.
4. Guo X., Xu M. Some physical applications of fractional Schrödinger equation. J. Math. Phys. 2006, 47:82-104.
5. Jeng M., Xu S.L.Y., Hawkins E., Schwarz J.M. On the nonlocality of the fractional Schrödinger equation. J. Math. Phys. 2010, 51:062102.
6. Bayin S.S. On the consistency of the solutions of the space fractional Schrödinger equation. J. Math. Phys. 2012, 53:042105.
7. Luchko Y. Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys. 2013, 54:012111.
8. Amore P., Fernández F.M., Hofmann C.P., Sáenz R.A. Collocation method for fractional quantum mechanics. J. Math. Phys. 2010, 51:122101.
9. 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.
10. Atangana A., Cloot A.H. Stability and convergence of the space fractional variable-order Schröinger equation. Adv. Differ. Equ. 2013, 2013:80.
11. Leble S., Reichel B. Coupled nonlinear Schrödinger equations in optic fibers theory: from general to solitonic aspects. Eur. Phys. J. Spec. Top. 2009, 173:5-55.
12. Guo B., Huo Z. Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg-Landau equation. Fract. Calc. Appl. Anal. 2013, 16(1):226-242.
13. Kirkpatrick K., Lenzmann E., Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions. Commun. Math. Phys. 2013, 317(3):563-591.
14. Secchi S., Squassina M. Soliton dynamics for fractional Schrodinger equations. Appl. Anal. 2013, 1-28. 10.1080/00036811.2013.844793.
15. RN. J. Math. Phys. 2013, 54:031501.
16. Cheng M. Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 2012, 53:043507.
17. Guo B., Han Y., Xin J. Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 2008, 204:468-477.
18. Hu J., Xin J., Lu H. The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition. Comput. Math. Appl. 2011, 62:1510-1521.
19. Wang T., Guo B., Zhang L. New conservative difference schemes for a coupled nonlinear Schrödinger system. Appl. Math. Comput. 2010, 217:1604-1619.
20. Zhou Y. Application of Discrete Functional Analysis to the Finite Difference Methods 1990, International Academic Publishers, Beijing.
21. L ∞ convergence of a difference secheme for coupled nonlinear Schrödinger equations. Comput. Math. Appl. 2010, 59:3286-3300.
22. Sun Z., Zhu Q. On Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation. J. Comput. Appl. Math. 1998, 98(2):289-304.
23. Ismail M.S., Taha T.R. A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation. Math. Comput. Simul. 2007, 74:302-311.
24. Bao W., Cai Y. Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation. Math. Comput. 2013, 82:99-128.
25. Bao W., Cai Y. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Models 2013, 6:1-135.
26. Bao W., Tang Q., Xu Z. Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation. J. Comput. Phys. 2013, 235:423-445.
27. Gauckler L., Lubich C. Nonlinear Schrödinger equations and their spectral discretizations over long times. Found. Comput. Math. 2010, 10:141-169.
28. Chen J.B., Qin M.Z., Tang Y.F. Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation. Comput. Math. Appl. 2002, 43(8-9):1095-1106.
29. Yang Q., Liu F., Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 2010, 34(1):200-218.
30. Shen S., Liu F., Anh V., Turner I., Chen J. A novel numerical approximation for the Riesz space fractional advection-dispersion equation. IMA J. Appl. Math. 2012, 10.1093/imamat/hxs073.
31. Ilić M., Liu F., Turner I., Anh V. Numerical approximation of a fractional-in-space diffusion equation, I. Fract. Calc. Appl. Anal. 2005, 8(3):323-341.
32. Ortigueira M.D. Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 1-12.
33. Çelik C., Duman M. Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 2012, 231:1743-1750.
34. Bradley C.C., Sackett C.A., Tollett J.J., Hulet R.G. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 1995, 75:1687-1690.
35. Caplan R.M., Carretero-Gonzalez R. A modulus-squared Dirichlet boundary condition for time-dependent complex partial differential equations and its application to the nonlinear Schrödinger equation. SIAM J. Sci. Comput. 2014, 36(1):1-19.
36. Demengel F., Demengel G. Fractional sobolev spaces. Functional Spaces for the Theory of Elliptic Partial Differential Equations 2012, 179-228. Springer, London.