Free Access
Volume 50, Number 4, July-August 2016
Page(s) 945 - 964
Published online 06 June 2016
  1. G. Agrawal, Nonlinear fiber optics. Academic Press, 4th edition (2006). [Google Scholar]
  2. L. Auslander and F.A. Grunbaum, The Fourier transform and the discrete Fourier transform. Inverse Probl. 5 (1989) 149–164. [CrossRef] [Google Scholar]
  3. S. Balac, High order Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method. J. KSIAM 17 (2013) 238–266. [Google Scholar]
  4. S. Balac and A. Fernandez, Mathematical analysis of adaptive step-size techniques when solving the nonlinear Schrödinger equation for simulating light-wave propagation in optical fibers. Opt. Commun. 329 (2014) 1–9. [CrossRef] [Google Scholar]
  5. S. Balac and F. Mahé, Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method. Comput. Phys. Commun. 184 (2013) 1211–1219. [CrossRef] [MathSciNet] [Google Scholar]
  6. C. Besse, B. Bidégaray and S. Descombes. Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. [CrossRef] [MathSciNet] [Google Scholar]
  7. J.C. Butcher, Numerical Methods for Ordinary Differential Equations. John Wiley and Sons (2008). [Google Scholar]
  8. B. Cano and A. González-Pachón, Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation. See (2013). [Google Scholar]
  9. B. Cano and A. González-Pachón, Exponential time integration of solitary waves of cubic Schrödinger equation. Appl. Numer. Math. 91 (2015) 26–45. [CrossRef] [MathSciNet] [Google Scholar]
  10. B.M. Caradoc−Davies. Vortex dynamics in Bose-Einstein condensate. Ph. D. thesis, University of Otago (NZ) (2000). [Google Scholar]
  11. R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations. World Scientific (2008). [Google Scholar]
  12. T. Cazenave, Semilinear Schrödinger Equations. Courant Lect. Notes Math. AMS, New York (2003). [Google Scholar]
  13. T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semi-linéaires. Ellipses, Paris (1990). [Google Scholar]
  14. E. Celledoni, D. Cohen and B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8 (2008) 303–317. [CrossRef] [MathSciNet] [Google Scholar]
  15. D. Cohen and L. Gauckler, One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52 (2012) 877–903. [CrossRef] [MathSciNet] [Google Scholar]
  16. S.M. Cox and P.C. Matthews, Exponential time-differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. [Google Scholar]
  17. M.J. Davis, Dynamics in Bose-Einstein condensate. Ph. D. thesis, University of Oxford (UK) (2001). [Google Scholar]
  18. J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6 (1980) 19–26. [CrossRef] [MathSciNet] [Google Scholar]
  19. G. Dujardin, Exponential Runge-Kutta methods for the Schrödinger equation. Appl. Numer. Math. 59 (2009) 1839–1857. [CrossRef] [MathSciNet] [Google Scholar]
  20. B.L. Ehle and J.D. Lawson, Generalized Runge-Kutta processes for stiff initial-value problems. J. Inst. Math. Appl. 16 (1975) 11–21. [CrossRef] [MathSciNet] [Google Scholar]
  21. C.L. Epstein. How well does the finite Fourier transform approximate the Fourier transform? Commun. Pure Appl. Math. 58 (2005) 1421–1435. [CrossRef] [Google Scholar]
  22. A. Fernandez, S. Balac, A. Mugnier, F. Mahé, R. Texier-Picard, T. Chartier and D. Pureur, Numerical simulation of incoherent optical wave propagation in nonlinear fibers. Eur. Phys. J. Appl. Phys. 64 (2013) 24506/1–11. [CrossRef] [EDP Sciences] [Google Scholar]
  23. M. Guenin, On the interaction picture. Commun. Math. Phys. 3 (1966) 120–132. [CrossRef] [Google Scholar]
  24. E. Hairer, S. P. Norsett and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag New York, Inc., New York, USA (1993). [Google Scholar]
  25. A. Heidt, Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers. J. Lightwave Technol. 27 (2009) 3984–3991. [CrossRef] [Google Scholar]
  26. M. Hochbruck and A. Ostermann, Exponential integrators. Acta Numer. 19 (2010) 209–286. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  27. J. Hult, A fourth-order Runge–Kutta in the Interaction Picture method for simulating supercontinuum generation in optical fibers. J. Lightwave Technol. 25 (2007) 3770–3775. [CrossRef] [Google Scholar]
  28. A.-K. Kassam and L.N. Trefethen. Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 (2005) 1214–1233. [CrossRef] [MathSciNet] [Google Scholar]
  29. J. D. Lawson, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4 (1967) 372–380. [CrossRef] [MathSciNet] [Google Scholar]
  30. C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–215. [Google Scholar]
  31. G.M. Muslu and H.A. Erbay, A split-step Fourier method for the complex modified Korteweg de Vries equation. Comput. Math. Appl. 45 (2003) 503–514. [CrossRef] [MathSciNet] [Google Scholar]
  32. O.V. Sinkin, R. Holzlöhner, J. Zweck and C.R. Menyuk, Optimization of the Split-Step Fourier method in modeling optical-fiber communications systems. J. Lightwave Technol. 21 (2003) 61. [CrossRef] [Google Scholar]
  33. M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258. [Google Scholar]
  34. J.S. Townsend, A modern approach to quantum mechanics. Internat. Series Pure Appl. Phys. University Science Books (2000). [Google Scholar]
  35. 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. [CrossRef] [MathSciNet] [Google Scholar]

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