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Journal of Computational Electronics

, Volume 3, Issue 1, pp 33–44 | Cite as

A Note on the Symplectic Integration of the Nonlinear Schrödinger Equation

  • Clemens Heitzinger
  • Christian Ringhofer
Article

Abstract

Numerically solving the nonlinear schröedinger equation and being able to treat arbitrary space dependent potentials permits many application in the realm of quantum mechanics. The long-term stability of a numerical method and its conservation properties is an important feature since it assures that the underlying physics of the solution are respected and it ensures that the numerical result is correct also for small time spans. In this paper we describe symplectic integrators for the nonlinear schröedinger equation with arbitrary potentials and perform numerical experiments comparing different approaches and highlighting their respective advantages and disadvantages.

nonlinear schröedinger equations symplectic integration difference methods 

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References

  1. 1.
    Semiconductor Industry Association, “International technology roadmap for semiconductors: Modeling and simulation” (2003). http://public.itrs.net.Google Scholar
  2. 2.
    C.S. Lent, P.D. Tougaw, W. Porod, and G.H. Bernstein, “Quantum cellular automata,” Nanotechnology, 4, 49 (1993).Google Scholar
  3. 3.
    C.S. Lent and P.D. Tougaw, “A device architecture for computing with quantum dots,” Proceedings of the IEEE, 85(4), 541 (1997).Google Scholar
  4. 4.
    D. Cai, D.W. McLaughlin, and K.T.R. McLaughlin, “The nonlinear Schrödinger equation as both a PDE and a dynamical system,” in Handbook of Dynamical Systems (North-Holland, Amsterdam, 2002), vol. 2, p. 599.Google Scholar
  5. 5.
    G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable opticalwave,” Phys. Rev. Lett., 87(3), 033902 (2001).Google Scholar
  6. 6.
    N.N. Akhmediev, “Nonlinear physics-déjà vu in optics,” Nature, 413, 267 (2001).Google Scholar
  7. 7.
    E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer-Verlag, Berlin, 2002).Google Scholar
  8. 8.
    V.A. Dorodnitsyn, “Transformation groups in net spaces,” J. Soviet Math., 55, 1490 (1991).Google Scholar
  9. 9.
    V.A. Dorodnitsyn, “Finite difference models entirely inheriting continuous symmetry of original differential equations,” Internat. J. Modern Phys. C, 5(4), 723 (1994).Google Scholar
  10. 10.
    V. Dorodnitsyn, “Noether-type theorems for difference equations,” Appl. Numer. Math., 39(3/4), 307 (2001).Google Scholar
  11. 11.
    H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Tome III (Gauthiers-Villars, Paris, 1899).Google Scholar
  12. 12.
    G. Benettin and A. Giorgilli, “On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms,” J. Statist. Phys., 74(5/6), 1117 (1994).Google Scholar
  13. 13.
    A. Clebsch, “Ueber die simultane integration linearer partieller differentialgleichungen,” Crelle Journal f. d. reine u. angew. Math., 65, 257 (1866).Google Scholar
  14. 14.
    G. Darboux, “Sur le problème de Pfaff,” C.R. XCIV. 835-837; Darb. Bull. (2) VI. 14, 49, 1882.Google Scholar
  15. 15.
    C.G.J. Jacobi, Gesammelte Werke (V. Band, G. Reimer, Berlin, 1890).Google Scholar
  16. 16.
    S. Lie, Gesammelte Abhandlungen, 5. Band: Abhandlungen über die Theorie der Transformationsgruppen (B. Teubner, Leipzig, 1924).Google Scholar
  17. 17.
    M.J. Ablowitz and J.F. Ladik, “A nonlinear difference scheme and inverse scattering,” Studies in Appl. Math., 55(3), 213 (1976).Google Scholar
  18. 18.
    B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1996).Google Scholar
  19. 19.
    Y.-F. Tang, V.M. Pérez-García, and L. Vázquez, “Symplectic methods for the Ablowitz-Ladik model,” Appl. Math. Comput., 82(1), 17 (1997).Google Scholar
  20. 20.
    E. Fermi, J. Pasta, and H.C. Ulam, in Collected Papers of Enrico Fermi, edited by E. Segrè (The University of Chicago, Chicago, 1965), vol. 2, pp. 977-988.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Clemens Heitzinger
    • 1
  • Christian Ringhofer
    • 1
  1. 1.Department of MathematicsArizona State UniversityTempeUSA

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