Copyright © 2012 Tingchun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

1. D. J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, Englewood Cliffs, NJ, USA, 1995.
2. C. R. Menyuk, “Stability of solitons in birefringent optical fibers,” Journal of the Optical Society of America B, vol. 5, pp. 392–402, 1998.
3. M. Wadati, T. Iizuka, and M. Hisakado, “A coupled nonlinear Schrödinger equation and optical solitons,” Journal of the Physical Society of Japan, vol. 61, no. 7, pp. 2241–2245, 1992. View at Publisher · View at Google Scholar
4. G. D. Akrivis, “Finite difference discretization of the cubic Schrödinger equation,” IMA Journal of Numerical Analysis, vol. 13, no. 1, pp. 115–124, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
5. T. F. Chan and L. J. Shen, “Stability analysis of difference schemes for variable coefficient Schrödinger type equations,” SIAM Journal on Numerical Analysis, vol. 24, no. 2, pp. 336–349, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
6. Q. Chang, E. Jia, and W. Sun, “Difference schemes for solving the generalized nonlinear Schrödinger equation,” Journal of Computational Physics, vol. 148, no. 2, pp. 397–415, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
7. W. Z. Dai, “An unconditionally stable three-level explicit difference scheme for the Schrödinger equation with a variable coefficient,” SIAM Journal on Numerical Analysis, vol. 29, no. 1, pp. 174–181, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
8. M. Dehghan and A. Taleei, “A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients,” Computer Physics Communications, vol. 181, no. 1, pp. 43–51, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
9. F. Ivanauskas and M. Radžiūnas, “On convergence and stability of the explicit difference method for solution of nonlinear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 36, no. 5, pp. 1466–1481, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
10. P. L. Nash and L. Y. Chen, “Efficient finite difference solutions to the time-dependent Schrödinger equation,” Journal of Computational Physics, vol. 130, no. 2, pp. 266–268, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
11. J. M. Sanz-Serna, “Methods for the numerical solution of the nonlinear Schroedinger equation,” Mathematics of Computation, vol. 43, no. 167, pp. 21–27, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
12. Z. Z. Sun and X. Wu, “The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions,” Journal of Computational Physics, vol. 214, no. 1, pp. 209–223, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
13. L. Wu, “Dufort-Frankel-type methods for linear and nonlinear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 33, no. 4, pp. 1526–1533, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
14. J. M. Sanz-Serna and J. G. Verwer, “Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation,” IMA Journal of Numerical Analysis, vol. 6, no. 1, pp. 25–42, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
15. Z. Fei, V. M. Pérez-García, and L. Vázquez, “Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme,” Applied Mathematics and Computation, vol. 71, no. 2-3, pp. 165–177, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
16. S. Li and L. Vu-Quoc, “Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1839–1875, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
17. G. D. Akrivis, V. A. Dougalis, and O. A. Karakashian, “On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation,” Numerische Mathematik, vol. 59, no. 1, pp. 31–53, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
18. O. Karakashian, G. D. Akrivis, and V. A. Dougalis, “On optimal order error estimates for the nonlinear Schrödinger equation,” SIAM Journal on Numerical Analysis, vol. 30, no. 2, pp. 377–400, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
19. Y. Tourigny, “Some pointwise estimates for the finite element solution of a radial nonlinear Schrödinger equation on a class of nonuniform grids,” Numerical Methods for Partial Differential Equations, vol. 10, no. 6, pp. 757–769, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
20. W. Bao and D. Jaksch, “An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1406–1426, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
21. T. Iitaka, “Solving the time-dependent Schröinger equation numerically,” Physical Review E, vol. 49, no. 5, pp. 4684–4690, 1994.
22. D. Kosloff and R. Kosloff, “A Fourier method solution for the time dependent Schröinger equation as a tool in molecular dynamics,” Journal of Computational Physics, vol. 52, pp. 35–53, 1983.
23. B. Li, G. Fairweather, and B. Bialecki, “Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables,” SIAM Journal on Numerical Analysis, vol. 35, no. 2, pp. 453–477, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
24. M. P. Robinson and G. Fairweather, “Orthogonal spline collocation methods for Schrödinger-type equations in one space variable,” Numerische Mathematik, vol. 68, no. 3, pp. 355–376, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
25. G. Berikelashvili, M. M. Gupta, and M. Mirianashvili, “Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations,” SIAM Journal on Numerical Analysis, vol. 45, no. 1, pp. 443–455, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
26. G. Cohen, High-Order Numerical Methods for Transient Wave Equations, Springer, New York, NY, USA, 2002.
27. A. Gopaul and M. Bhuruth, “Analysis of a fourth-order scheme for a three-dimensional convection-diffusion model problem,” SIAM Journal on Scientific Computing, vol. 28, no. 6, pp. 2075–2094, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
28. B. Gustafsson and E. Mossberg, “Time compact high order difference methods for wave propagation,” SIAM Journal on Scientific Computing, vol. 26, no. 1, pp. 259–271, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
29. B. Gustafsson and P. Wahlund, “Time compact difference methods for wave propagation in discontinuous media,” SIAM Journal on Scientific Computing, vol. 26, no. 1, pp. 272–293, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
30. J. C. Kalita, D. C. Dalal, and A. K. Dass, “A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients,” International Journal for Numerical Methods in Fluids, vol. 38, no. 12, pp. 1111–1131, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
31. S. K. Lele, “Compact finite difference schemes with spectral-like resolution,” Journal of Computational Physics, vol. 103, no. 1, pp. 16–42, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
32. H. L. Liao and Z. Z. Sun, “Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 37–60, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
33. H. L. Liao, Z. Z. Sun, and H. S. Shi, “Error estimate of fourth-order compact scheme for linear Schrödinger equations,” SIAM Journal on Numerical Analysis, vol. 47, no. 6, pp. 4381–4401, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
34. S. Xie, G. Li, and S. Yi, “Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 9-12, pp. 1052–1060, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
35. D. W. Zingg, “Comparison of high-accuracy finite-difference methods for linear wave propagation,” SIAM Journal on Scientific Computing, vol. 22, no. 2, pp. 476–502, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
36. R. A. Horn and C. R. Johnson, Matrix Analysis, chapter 7, Cambridge University Press, 1985.
37. F. E. Browder, “Existence and uniqueness theorems for solutions of nonlinear boundary value problems,” in Application of Nonlinear Partial Differential Equations, R. Finn, Ed., Proceedings of Symposia in Applied Mathematics, pp. 24–49, American Mathematical Society, Providence, RI, USA, 1965.