Numerical solution of nonlinear schrödinger equation by using time-space pseudo-spectral method

Numerical Methods for Partial Differential Equations

Volumn 26, Issue 4, 2010, Pages 979-992

  Dehghan, Mehdi ^a   Taleei, Ameneh ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Chebyshev Gauss Lobbato (CGL) quadrature points; Fourier pseudo spectral time splitting (FPTS) method; Gross Pitaveskii (GP) equation; Nonlinear Schr dinger equation (NLS); Pseudo spectral method; Soliton; Time space pseudo spectral (TSPS)method

Indexed keywords

CHEBYSHEV; DINGER EQUATION; FOURIER; PSEUDO-SPECTRAL METHOD; PSEUDOSPECTRAL METHODS; TIME SPLITTING; TIME-SPACE;

NUMERICAL ANALYSIS; SOLITONS; SPECTROSCOPY;

NONLINEAR EQUATIONS;

EID: 77952893917     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20468     Document Type: Article

References (52)

1. G. P. Agrawal, Nonlinear fiber optics, 3rd Ed., Academic Press, San Diego, 2001.
2. A. P. Fordy, Soliton theory: a survey of results, Manchester University Press, Manchester, 1990.
3. L. Stenflo and M. Y. Yu, Nonlinear wave modulation in a cylindrical plasma, IEEE Trans Plasma Sci 25 (1997), 1155-1157.
4. C. Sulem and P. L. Sulem, The nonlinear Schrodinger equation: self-focusing and wave collapse, Springer, New York, 1999.
5. J. A. C. Weideman and B. M. Herbst, Split-step methods for the solution of the nonlinear Schrodinger equation, SIAM J Numer Anal 23 (1986), 485-507.
6. I. Dag, A quadratic B-spline finite element method for solving nonlinear Schrodinger equation, Comput Methods Appl Mech Eng 174 (1999), 247-258.
7. D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrodinger equations, J Comput Phys 87 (1990), 108-125.
8. W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J Comput Phys 187 (2003), 318-342.
9. W. Bao and J. Shen, A fourth-order time-splitting Lagurre-Hermite pseudo-spectral method for Bose-Einstein condensates, SIAM J Sci Comput 26 (2005), 2010-2028.
10. J. Shen and L. L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl Numer Math 57 (2007), 710-720.
11. S. Yu, S. Zhao, and G. W. Wei, Local spectral time splitting method for first and second-order partial differential equations, J Comput Phys 206 (2005), 727-780.
12. M. Delfour, M. Fortin, and G. Payre, Finite difference solutions of a nonlinear Schrodinger equation, J Comput Phys 44 (1981), 277-288.
13. S. R. K. Iyengar, G. Jayaraman, and V. Balasubramanian, Variable mesh difference schemes for solving a nonlinear Schrodinger equation with a linear damping term, Comput Math Appl 40 (2000), 1375-1385.
14. H. Q. Wang, Numerical studies on the split-step finite difference method for nonlinear Schrodinger equations, Appl Math Comput 170 (2005), 17-35.
15. M. Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math Comput Simul 71 (2006), 16-30.
16. J. P. Boyd, Chebyshev and Fourier spectral methods, Springer-Verlag (Berlin, New York), 1989. 17. P. Muruganandam and S. K. Adhikari, Bose-Einstein condensation dynamics in three dimensions by the pseudo-spectral and finite difference methods, J Phys B 36 (2003), 2501-2513.
17. W. Bao, S. Jin, and P. A. Markowich, On time-splitting spectral approximation for the Schrodinger equation in the semiclassical regime, J Comput Phys 175 (2002), 487-524.
18. B. Fornberg, A practical guide to pseudo-spectral methods, Cambridge University Press, 1996.
19. B. Fornberg and T. A. Driscoll, A fast spectral algorithm for nonlinear wave equation with linear dispersion, J Comput Phys 155 (1999), 456-467.
20. M. Javidi and A. Golbabai, Numerical studies on nonlinear Schrodinger equations by spectral collocation method with preconditioning, J Math Anal Appl 333 (2007), 1119-1127.
21. M. Dehghan and M. Shamsi, Numerical solution of two-dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral Legendre method, Numer Methods Partial Differential Eq 22 (2006), 1255-1266.
22. G. Delic, Spectral function methods for nonlinear diffusion equations, J Math Phys 28 (1987), 39-59.
23. M. Deville, L. Kleiser, and F. Montigny-Rannou, Pressure and time treatment for Chebyshev spectral solution of a Stokes problem, Int J Numer Meth Fluids 4 (1984), 1149-1163.
24. P. Dutt, Spectral methods for initial boundary value problems, an alternative approach, SIAM J Numer Anal 27 (1990), 885-903.
25. D. L. Lewis, J. Lund, and K. L. Bowers, The space-time sinc-Galerkin method for parabolic problems, Int J Num Methods Eng 24 (1987), 1629-1644.
26. G. Ierley, B. J. Spencer, and R. A. Worthing, Spectral methods in time for a class of parabolic partial differential equations, J Comput Phys 102 (1992), 88-97.
27. Y. Morchoisne, Pseudo-spectral space-time calculations of incompressible viscous flows, AIAA 81 (1981), 81-109.
28. Y. Morchoisne, Resolution of Navier-Stokes equations by a space-time pseudospectral method, Rech Aerosp 5 (1979), 293-306.
29. Q. S. Chang, E. H. Jia, and W. Sun, Difference schemes for solving the generalized nonlinear Schrodinger equation, J Comput Phys 148 (1999), 397-415.
30. M. M. Cerimele, M. L. Chiofalo, F. Pistella, S. Succi, and M. P. Tosi, Numerical solution of the Gross-Pitaevskii equation using an explicit finite difference scheme: an application to trapped Bose-Einstein condensates, Phys Rev E 62 (2000), 1382-1389.
31. J. B. Chen, M. Z. Qin, and Y. F. Tang, Symplectic and multi-symplectic methods for the nonlinear Schrodinger equation, Comput Math Appl 43 (2002), 1095-1106.
32. W. Dai, Absolute stable explicit and semi-explicit schemes for Schrodinger equations, Math Numer Sinica 11 (1989), 128-131.
33. F. Ivanauskas and M. Radziunas, On convergence and stability of the explicit difference method for solution of nonlinear Schrodinger equations, SIAM J Numer Anal 36 (1999), 1466-1481.
34. M. C. Lai and C. Y. Huang, A simple Dufort-Frankel type scheme for the Gross-pitaevskii equation of Bose-Einstein condensates on different geometries, Numer Methods Partial Differential Eq 20 (2004), 628-638.
35. G. M. Muslu and H. A. Erbay, High-order split-step Fourier schemes for the generalized nonlinear Schrodinger equation, Math Comput Simul 67 (2005), 581-592.
36. H. P. Pfeiffer, L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, A multidomain spectral method for solving elliptic equations, Comput Phys Commun 152 (2003), 253-273.
37. J. I. Ramos, Linearly implicit methods for the nonlinear Schrodinger equation in nonhomogeneous media, Appl Math Comput 133 (2002), 1-28.
38. S. Reich, Multi-symplectic Runge-Kutta methods for Hamiltonian wave equations, J Comput Phys 157 (2000), 473-499.
39. L. Wu, Dufort-Frankel type methods for linear and nonlinear Schrodinger equations, SIAM J Numer Anal 33 (1996), 1526-1533.
40. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, New York, 1988.
41. M. Shamsi and M. Dehghan, Recovering a time-dependent coefficient in a parabolic equation from over-specified boundary data using the pseudo-spectral Legendre method, Numer Methods Partial Differential Eq 23 (2007), 196-210.
42. R. Baltensperger and M. R. Trummer, Spectral differencing with a twist, SIAM J Sci Comput 24 (2003), 1465-1487.
43. B. D. Welfert, Generation of pseudo-spectral differentiation, SIAM J Numer Anal 34 (1997), 1640-1657.
44. L. N. Trefethen, Spectral methods in Matlab, SIAM, Philadelphia, 2000.
45. J. G. Tang and H. P. Ma, Single and multi-interval spectral methods in time for parabolic equations, Numer Methods Partial Differen Equations 22 (2006), 1007-1034.
46. M. Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods Partial Differential Eq 21 (2005), 24-40.
47. M. Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math Comput Model 41 (2005), 196-213.
48. M. Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Differential Eq 22 (2006), 220-257.
49. M. Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos, Solitons Fractals 32 (2007), 661-675.
50. M. Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Nonlinear Sciences Numer Simul 7 (2006), 447-450.
51. M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl Numer Math 52 (2005), 39-62.
52. M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer Methods Partial Differential Eq 21 (2005), 611-622. (Pubitemid 40668509)