Stability of the higher-order splitting methods for the nonlinear Schrödinger equation with an arbitrary dispersion operator
Abstract
The numerical solution of the generalized nonlinear Schrödinger equation by simple splitting methods can be disturbed by so-called spurious instabilities. We analyze these numerical instabilities for an arbitrary splitting method and apply our results to several well-known higher-order splittings. We find that the spurious instabilities can be suppressed to a large extent. However, they never disappear completely if one keeps the integration step above a certain limit and applies what is considered to be a more accurate higher-order method. The latter can be used to make calculations more accurate with the same numerically stable step, but not to make calculations faster with a much larger step.
Keyword : nonlinear optics, nonlinear fibers, nonlinear Schrödinger equation, generalized nonlinear Schrödinger equation (GNLSE), modulation instability (MI), four-wave mixing, spurious instabilities, splitting methods
How to Cite
Amiranashvili, S., & Čiegis, R. (2024). Stability of the higher-order splitting methods for the nonlinear Schrödinger equation with an arbitrary dispersion operator. Mathematical Modelling and Analysis, 29(3), 560–574. https://doi.org/10.3846/mma.2024.20905
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
G. P. Agrawal. Nonlinear Fiber Optics. Academic, New York, 4 edition, 2007.
Sh. Amiranashvili, M. Radziunas, U. Bandelow, K. Busch and R. Čiegis. Additive splitting methods for parallel solutions of evolution problems. Journal of Computational Physics, 436(110320):1–14, July 2021. https://doi.org/10.1016/j.jcp.2021.110320
W. Auzinger, H. Hofstätter and O. Koch. Coefficients of various splitting methods. https://www.asc.tuwien.ac.at/~winfried/splitting/
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe (Eds.). Handbook of Optics, volume 1. McGRAW-HILL, 2 edition, 1995.
A. Blanco-Redondo, C.M. de Sterke, J.E. Sipe, T.F. Krauss, B.J. Eggleton and C. Husko. Pure-quartic solitons. Nat. Commun., 7(10427):1–9, January 2016. https://doi.org/10.1038/ncomms10427
S. Blanes and P.C. Moan. Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. Journal of Computational and Applied Mathematics, 142(2):313–330, May 2002. https://doi.org/10.1016/S0377-0427(01)00492-7
G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini and S. Benedetto. Suppression of spurious tones induced by the split-step method in fiber systems simulation. IEEE Photonics Technology Letters, 13(5):489–491, May 2000. https://doi.org/10.1109/68.841262
R. W. Boyd. Nonlinear Optics. Academic, New York, 3 edition, 2008.
J. M. Dudley, G. Genty and S. Coen. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys., 78(4):1135–1184, 2006. https://doi.org/10.1103/RevModPhys.78.1135
Roland Glowinski, Stanley J. Osher and Wotao Yin (Eds.). Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Berlin, 2016.
J.P. Gordon. Theory of the soliton self-frequency shift. Opt. Lett., 11(10):662– 664, October 1986. https://doi.org/10.1364/OL.11.000662
V. I. Karpman. Non-linear waves in dispersive media. Pergamon, 1975.
E. Kartashova. Nonlinear Resonance Analysis. Cambridge University Press, 2010.
T.I. Lakoba. Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation. Numerical Methods for Partial Differential Equations, 28(2):641–669, March 2012. https://doi.org/10.1002/num.20649
T.I. Lakoba. Instability of the split-step method for a signal with nonzero central frequency. J. Opt. Soc. Am. B, 30(12):3260–3271, December 2013. https://doi.org/10.1364/JOSAB.30.003260
T.I. Lakoba. Long-time simulations of nonlinear Schrödinger-type equations using step size exceeding threshold of numerical instability. J Sci Comput, 72(1):14–48, July 2017. https://doi.org/10.1007/s10915-016-0346-y
Martino Lovisetto, Didier Clamond and Bruno Marcos. Integrating factor techniques applied to the Schrödinger-like equations. Comparison with SplitStep methods. Applied Numerical Mathematics, 197:258–271, March 2024. https://doi.org/10.1016/j.apnum.2023.11.016
R.I. McLachlan and R. Quispel. Splitting methods. Acta Numerica, 11:341–434, January 2002. https://doi.org/10.1017/S0962492902000053
F.M. Mitschke and L.F. Mollenauer. Discovery of the soliton self-frequency shift. Opt. Lett., 11(10):569–661, 1986. https://doi.org/10.1364/OL.11.000659
A. H. Nayfeh. Perturbation methods. Wiley, 1973.
F. Severing, U. Bandelow and Sh. Amiranashvili. Spurious four-wave mixing processes in generalized nonlinear Schro¨dinger equations. Preprint 2975, WIAS, Mohrenstr. 39, 10117 Berlin, November 2022. Available on Internet: https://www.wias-berlin.de/preprint/2975/wias_preprints_2975.pdf
F. Severing, U. Bandelow and Sh. Amiranashvili. Spurious four-wave mixing processes in generalized nonlinear Schrödinger equations. J. Lightwave Technol., 41(16):5359–5365, August 2023. https://doi.org/10.1109/JLT.2023.3261804
M. Suzuki. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A, 146(6):319– 323, June 1990. https://doi.org/10.1016/0375-9601(90)90962-N
T.R. Taha and M.I. Ablowitz. Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. Journal of Computational Physics, 55(2):231–253, August 1984. https://doi.org/10.1016/0021-9991(84)90003-2
J.A.C. Weideman and B.M. Herbst. Split-Step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal., 23(3):485–507, June 1986. https://doi.org/10.1016/0021-9991(84)90003-2
G. B. Whitham. Linear and nonlinear waves. John Wiley & Sons, New York, 1974.
J. Yang. Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, 2010.
H. Yoshida. Construction of higher order symplectic integrators. Phys. Lett. A, 150(5-7):262–268, November 1990. https://doi.org/10.1017/S0074180900091440
V.E. Zakharov and L.A. Ostrovsky. Modulation instability: the beginning. Physica D: Nonlinear Phenomena, 238(5):540–548, March 2009. https://doi.org/10.1016/j.physd.2008.12.002
V.E. Zakharov and A.B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 34(1):62–69, 1972.
Sh. Amiranashvili, M. Radziunas, U. Bandelow, K. Busch and R. Čiegis. Additive splitting methods for parallel solutions of evolution problems. Journal of Computational Physics, 436(110320):1–14, July 2021. https://doi.org/10.1016/j.jcp.2021.110320
W. Auzinger, H. Hofstätter and O. Koch. Coefficients of various splitting methods. https://www.asc.tuwien.ac.at/~winfried/splitting/
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe (Eds.). Handbook of Optics, volume 1. McGRAW-HILL, 2 edition, 1995.
A. Blanco-Redondo, C.M. de Sterke, J.E. Sipe, T.F. Krauss, B.J. Eggleton and C. Husko. Pure-quartic solitons. Nat. Commun., 7(10427):1–9, January 2016. https://doi.org/10.1038/ncomms10427
S. Blanes and P.C. Moan. Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. Journal of Computational and Applied Mathematics, 142(2):313–330, May 2002. https://doi.org/10.1016/S0377-0427(01)00492-7
G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini and S. Benedetto. Suppression of spurious tones induced by the split-step method in fiber systems simulation. IEEE Photonics Technology Letters, 13(5):489–491, May 2000. https://doi.org/10.1109/68.841262
R. W. Boyd. Nonlinear Optics. Academic, New York, 3 edition, 2008.
J. M. Dudley, G. Genty and S. Coen. Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys., 78(4):1135–1184, 2006. https://doi.org/10.1103/RevModPhys.78.1135
Roland Glowinski, Stanley J. Osher and Wotao Yin (Eds.). Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Berlin, 2016.
J.P. Gordon. Theory of the soliton self-frequency shift. Opt. Lett., 11(10):662– 664, October 1986. https://doi.org/10.1364/OL.11.000662
V. I. Karpman. Non-linear waves in dispersive media. Pergamon, 1975.
E. Kartashova. Nonlinear Resonance Analysis. Cambridge University Press, 2010.
T.I. Lakoba. Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation. Numerical Methods for Partial Differential Equations, 28(2):641–669, March 2012. https://doi.org/10.1002/num.20649
T.I. Lakoba. Instability of the split-step method for a signal with nonzero central frequency. J. Opt. Soc. Am. B, 30(12):3260–3271, December 2013. https://doi.org/10.1364/JOSAB.30.003260
T.I. Lakoba. Long-time simulations of nonlinear Schrödinger-type equations using step size exceeding threshold of numerical instability. J Sci Comput, 72(1):14–48, July 2017. https://doi.org/10.1007/s10915-016-0346-y
Martino Lovisetto, Didier Clamond and Bruno Marcos. Integrating factor techniques applied to the Schrödinger-like equations. Comparison with SplitStep methods. Applied Numerical Mathematics, 197:258–271, March 2024. https://doi.org/10.1016/j.apnum.2023.11.016
R.I. McLachlan and R. Quispel. Splitting methods. Acta Numerica, 11:341–434, January 2002. https://doi.org/10.1017/S0962492902000053
F.M. Mitschke and L.F. Mollenauer. Discovery of the soliton self-frequency shift. Opt. Lett., 11(10):569–661, 1986. https://doi.org/10.1364/OL.11.000659
A. H. Nayfeh. Perturbation methods. Wiley, 1973.
F. Severing, U. Bandelow and Sh. Amiranashvili. Spurious four-wave mixing processes in generalized nonlinear Schro¨dinger equations. Preprint 2975, WIAS, Mohrenstr. 39, 10117 Berlin, November 2022. Available on Internet: https://www.wias-berlin.de/preprint/2975/wias_preprints_2975.pdf
F. Severing, U. Bandelow and Sh. Amiranashvili. Spurious four-wave mixing processes in generalized nonlinear Schrödinger equations. J. Lightwave Technol., 41(16):5359–5365, August 2023. https://doi.org/10.1109/JLT.2023.3261804
M. Suzuki. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A, 146(6):319– 323, June 1990. https://doi.org/10.1016/0375-9601(90)90962-N
T.R. Taha and M.I. Ablowitz. Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation. Journal of Computational Physics, 55(2):231–253, August 1984. https://doi.org/10.1016/0021-9991(84)90003-2
J.A.C. Weideman and B.M. Herbst. Split-Step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal., 23(3):485–507, June 1986. https://doi.org/10.1016/0021-9991(84)90003-2
G. B. Whitham. Linear and nonlinear waves. John Wiley & Sons, New York, 1974.
J. Yang. Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, 2010.
H. Yoshida. Construction of higher order symplectic integrators. Phys. Lett. A, 150(5-7):262–268, November 1990. https://doi.org/10.1017/S0074180900091440
V.E. Zakharov and L.A. Ostrovsky. Modulation instability: the beginning. Physica D: Nonlinear Phenomena, 238(5):540–548, March 2009. https://doi.org/10.1016/j.physd.2008.12.002
V.E. Zakharov and A.B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 34(1):62–69, 1972.