The “interaction picture” (IP) method is a very promising alternative to Split-Step methods for solving certain type of partial differential equations such as the nonlinear Schrödinger equation used in the simulation of wave propagation in optical fibers. The method exhibits interesting convergence properties and is likely to provide more accurate numerical results than cost comparable Split-Step methods such as the Symmetric Split-Step method. In this work we investigate in detail the numerical properties of the IP method and carry out a precise comparison between the IP method and the Symmetric Split-Step method.
DOI : 10.1051/m2an/2015060
Mots-clés : Interaction picture method, symmetric Split-Step method, Runge−Kutta method, nonlinear optics, nonlinear Schrödinger equation
@article{M2AN_2016__50_4_945_0,
author = {Balac, St\'ephane and Fernandez, Arnaud and Mah\'e, Fabrice and M\'ehats, Florian and Texier-Picard, Rozenn},
title = {The {Interaction} {Picture} method for solving the generalized nonlinear {Schr\"odinger} equation in optics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {945--964},
publisher = {EDP-Sciences},
volume = {50},
number = {4},
year = {2016},
doi = {10.1051/m2an/2015060},
zbl = {1401.78014},
mrnumber = {3521707},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2015060/}
}
TY - JOUR AU - Balac, Stéphane AU - Fernandez, Arnaud AU - Mahé, Fabrice AU - Méhats, Florian AU - Texier-Picard, Rozenn TI - The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 945 EP - 964 VL - 50 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015060/ DO - 10.1051/m2an/2015060 LA - en ID - M2AN_2016__50_4_945_0 ER -
%0 Journal Article %A Balac, Stéphane %A Fernandez, Arnaud %A Mahé, Fabrice %A Méhats, Florian %A Texier-Picard, Rozenn %T The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 945-964 %V 50 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015060/ %R 10.1051/m2an/2015060 %G en %F M2AN_2016__50_4_945_0
Balac, Stéphane; Fernandez, Arnaud; Mahé, Fabrice; Méhats, Florian; Texier-Picard, Rozenn. The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 945-964. doi : 10.1051/m2an/2015060. http://www.numdam.org/articles/10.1051/m2an/2015060/
G. Agrawal, Nonlinear fiber optics. Academic Press, 4th edition (2006).
and , The Fourier transform and the discrete Fourier transform. Inverse Probl. 5 (1989) 149–164. | DOI | MR | Zbl
, High order Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method. J. KSIAM 17 (2013) 238–266. | MR | Zbl
and , 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. | DOI
and , Embedded Runge-Kutta scheme for step-size control in the Interaction Picture method. Comput. Phys. Commun. 184 (2013) 1211–1219. | DOI | MR | Zbl
, and . Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002) 26–40. | DOI | MR | Zbl
J.C. Butcher, Numerical Methods for Ordinary Differential Equations. John Wiley and Sons (2008). | MR | Zbl
B. Cano and A. González-Pachón, Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation. See http://hermite.mac.cie.uva.es/bego/cgp3.pdf (2013).
and , Exponential time integration of solitary waves of cubic Schrödinger equation. Appl. Numer. Math. 91 (2015) 26–45. | DOI | MR | Zbl
B.M. Caradoc−Davies. Vortex dynamics in Bose-Einstein condensate. Ph. D. thesis, University of Otago (NZ) (2000).
R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations. World Scientific (2008). | MR | Zbl
T. Cazenave, Semilinear Schrödinger Equations. Courant Lect. Notes Math. AMS, New York (2003). | MR | Zbl
T. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semi-linéaires. Ellipses, Paris (1990). | MR | Zbl
, and , Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8 (2008) 303–317. | DOI | MR | Zbl
and , One-stage exponential integrators for nonlinear Schrödinger equations over long times. BIT 52 (2012) 877–903. | DOI | MR | Zbl
and , Exponential time-differencing for stiff systems. J. Comput. Phys. 176 (2002) 430–455. | DOI | MR | Zbl
M.J. Davis, Dynamics in Bose-Einstein condensate. Ph. D. thesis, University of Oxford (UK) (2001).
and , A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6 (1980) 19–26. | DOI | MR | Zbl
, Exponential Runge-Kutta methods for the Schrödinger equation. Appl. Numer. Math. 59 (2009) 1839–1857. | DOI | MR | Zbl
and , Generalized Runge-Kutta processes for stiff initial-value problems. J. Inst. Math. Appl. 16 (1975) 11–21. | DOI | MR | Zbl
. How well does the finite Fourier transform approximate the Fourier transform? Commun. Pure Appl. Math. 58 (2005) 1421–1435. | DOI | MR | Zbl
, , , , , and , Numerical simulation of incoherent optical wave propagation in nonlinear fibers. Eur. Phys. J. Appl. Phys. 64 (2013) 24506/1–11. | DOI
, On the interaction picture. Commun. Math. Phys. 3 (1966) 120–132. | DOI | MR | Zbl
E. Hairer, S. P. Norsett and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag New York, Inc., New York, USA (1993). | MR | Zbl
, Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers. J. Lightwave Technol. 27 (2009) 3984–3991. | DOI
and , Exponential integrators. Acta Numer. 19 (2010) 209–286. | DOI | MR | Zbl
, A fourth-order Runge–Kutta in the Interaction Picture method for simulating supercontinuum generation in optical fibers. J. Lightwave Technol. 25 (2007) 3770–3775. | DOI
and . Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 (2005) 1214–1233. | DOI | MR | Zbl
, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4 (1967) 372–380. | DOI | MR | Zbl
, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008) 2141–215. | DOI | MR | Zbl
and , A split-step Fourier method for the complex modified Korteweg de Vries equation. Comput. Math. Appl. 45 (2003) 503–514. | DOI | MR | Zbl
, , and , Optimization of the Split-Step Fourier method in modeling optical-fiber communications systems. J. Lightwave Technol. 21 (2003) 61. | DOI
, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations. SIAM J. Numer. Anal. 50 (2012) 3231–3258. | DOI | MR | Zbl
J.S. Townsend, A modern approach to quantum mechanics. Internat. Series Pure Appl. Phys. University Science Books (2000).
and , Split-step methods for the solution of the nonlinear Schrodinger equation. SIAM J. Numer. Anal. 23 (1986) 485–507. | DOI | MR | Zbl
Cité par Sources :
