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Fast evaluation of exact transparent boundary condition for one-dimensional cubic nonlinear Schrödinger equation

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Abstract

Fast evaluation of the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation is considered in this paper. In [J. Comput. Math., 2007, 25(6): 730–745], the author proposed a fast evaluation method for the half-order time derivative operator. In this paper, we apply this method for the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation. Numerical tests demonstrate the effectiveness of the proposed method.

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References

  1. Antoine X, Arnold A, Besse C, Ehrhardt M, Schädle A. A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Comm Comput Sci, 2008, 4: 729–796

    Google Scholar 

  2. Antoine X, Besse C. Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation. J Comput Phys, 2003, 181(1): 157–175

    Article  MathSciNet  Google Scholar 

  3. Antoine X, Besse C, Descombes S. Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations, SIAM J Numer Anal, 2006, 43(6): 2272–2293

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnold A. Numerically absorbing boundary conditions for quantum evolution equations. VLSI Design, 1998, 6: 313–319

    Article  Google Scholar 

  5. Arnold A, Ehrhardt M. Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics. J Comput Phys, 1998, 145: 611–638

    Article  MATH  MathSciNet  Google Scholar 

  6. Arnold A, Ehrhardt M, Sofronov I. Approximation, stability and fast calculation of non-local boundary conditions for the Schrödinger equation. Commun Math Sci, 2003, 1(3): 501–556

    MATH  MathSciNet  Google Scholar 

  7. Baskakov V A, Popov A V. Implementation of transparent boundaries for the numerical solution of the Schrödinger equation. Wave Motion, 1991, 14: 123–128

    Article  MathSciNet  Google Scholar 

  8. Bell W W. Special Functions for Scientists and Engineers. New York: D Van Nostrand Company, Ltd, 1968

    MATH  Google Scholar 

  9. Besse C. A relaxation scheme for the nonlinear Schrödinger equation. SIAM J Numer Anal, 2004, 42(3): 934–952

    Article  MATH  MathSciNet  Google Scholar 

  10. Boutet de Monvel A, Fokas A S, Shepelsky D. Analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett Math Phys, 2003, 65: 199–212

    Article  MATH  MathSciNet  Google Scholar 

  11. Davydov A S. Solitons in Molecular Systems. Dordrecht: Reidel, 1985

    MATH  Google Scholar 

  12. Delfour M, Fortin M, Payre G. Finite-difference solutions of a nonlinear Schrödinger equation. J Comput Phys, 1981, 44: 277–288

    Article  MATH  MathSciNet  Google Scholar 

  13. Durán A, Sanz-Serna J M. The numerical integration of relative equilibrium solution. The nonlinear Schrödinger equation. IMA J Numer Anal, 2000, 20(2): 235–261

    Article  MATH  MathSciNet  Google Scholar 

  14. Ehrhardt M, Arnold A. Discrete transparent boundary conditions for the Schrödinger equation. Riv Math Univ Parma, 2001, 6(4): 57–108

    MathSciNet  Google Scholar 

  15. Gorenflo R, Mainardi F. Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A, Mainardi F, Eds. Fractals and Fractional Calculus in Continuum Mechanics. Wien: Springer, 1997, 223–276

    Google Scholar 

  16. Han H, Huang Z Y. Exact artificial boundary conditions for Schrödinger equation in ℝ2. Comm Math Sci, 2004, 2: 79–94

    MATH  MathSciNet  Google Scholar 

  17. Han H, Yin D S, Huang Z Y. Numerical solutions of Schrödinger equations in ℝ3. Numer Meth Partial Diff Eqs, 2006, 23: 511–533

    Article  MathSciNet  Google Scholar 

  18. Hasegawa A. Optical Solitons in Fibers. Berlin: Springer-Verlag, 1989

    Google Scholar 

  19. Jiang S, Greengard L. Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comput Math Appl, 2004, 47: 955–966

    Article  MATH  MathSciNet  Google Scholar 

  20. Mayfield B. Non Local Boundary Conditions for the Schrödinger Equation. Ph D Thesis. Providence: University of Rhodes Island, 1989

    Google Scholar 

  21. Papadakis J S. Exact nonreflecting boundary conditions for parabolic-type approximations in underwater acoustics. J Comput Acoust, 1994, 2(2): 83–98

    Article  Google Scholar 

  22. Schmidt F, Deuflhard P. Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation. Comput Math Appl, 1995, 29(9): 53–76

    Article  MATH  MathSciNet  Google Scholar 

  23. Schmidt F, Yevick D. Discrete transparent boundary conditions for Schrödinger-type equations. J Comput Phys, 1997, 134: 96–107

    Article  MATH  MathSciNet  Google Scholar 

  24. Sulem C, Sulem P. The Nonlinear Schrödinger Equation. Berlin: Springer, 2000

    Google Scholar 

  25. Sun Z Z, Wu X. The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions. J Comput Phys, 2006, 214(1): 209–223

    Article  MATH  MathSciNet  Google Scholar 

  26. Yevick D, Friese T, Schmidt F. A comparison of transparent boundary conditions for the Fresnel equation. J Comput Phys, 2001, 168: 433–444

    Article  MATH  MathSciNet  Google Scholar 

  27. Zheng C. Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations. J Comput Phys, 2006, 215: 552–565

    Article  MATH  MathSciNet  Google Scholar 

  28. Zheng C. Approximation, stability and fast evaluation of exact artificial boundary condition for the one-dimensional heat equation. J Comput Math, 2007, 25(6): 730–745

    MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Chunxiong Zheng

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  1. Chunxiong Zheng
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Correspondence to Chunxiong Zheng.

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Zheng, C. Fast evaluation of exact transparent boundary condition for one-dimensional cubic nonlinear Schrödinger equation. Front. Math. China 5, 589–606 (2010). https://doi.org/10.1007/s11464-010-0058-9

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  • DOI: https://doi.org/10.1007/s11464-010-0058-9

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