Numerische Mathematik

, Volume 138, Issue 4, pp 975–1009 | Cite as

Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation

  • Tobias Jahnke
  • Marcel MiklEmail author


The dispersion-managed nonlinear Schrödinger equation contains a small parameter \(\varepsilon \), a rapidly changing piecewise constant coefficient function, and a cubic nonlinearity. Typical solutions are highly oscillatory and have a discontinuous time-derivative, and hence solving this equation numerically is a challenging task. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes.

Mathematics Subject Classification

65M12 65M15 65M70 65Z05 35B40 35Q55 



We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Moreover, we thank the anonymous referee for many helpful comments.


  1. 1.
    Abdulle, A., Weinan, E., Engquist, B., Vanden-Eijnden, E.: The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ablowitz, M.J., Biondini, G.: Multiscale pulse dynamics in communication systems with strong dispersion management. Opt. Lett. 23(21), 1668–1670 (1998)CrossRefGoogle Scholar
  3. 3.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic, Oxford (2013)zbMATHGoogle Scholar
  4. 4.
    Auzinger, W., Kassebacher, T., Koch, O., Thalhammer, M.: Adaptive splitting methods for nonlinear Schrödinger equations in the semiclassical regime. Numer. Algorithms 72(1), 1–35 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bader, P., Iserles, A., Kropielnicka, K., Singh, P.: Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2193), 20150733 (2016)CrossRefzbMATHGoogle Scholar
  6. 6.
    Baumstark, S., Faou, E., Schratz, K.: Uniformly accurate exponential-type integrators for Klein–Gordon equations with asymptotic convergence to the classical NLS splitting. CRC 1173-Preprint 2017/1, Karlsruhe Institute of Technology (2017). To appear in Math. Comput
  7. 7.
    Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society, Providence (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Calvo, M., Chartier, P., Murua, A., Sanz-Serna, J.: Numerical stroboscopic averaging for ODEs and DAEs. Appl. Numer. Math. 61(10), 1077–1095 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Castella, F., Chartier, P., Méhats, F., Murua, A.: Stroboscopic averaging for the nonlinear Schrödinger equation. Found. Comput. Math. 15(2), 519–559 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chartier, P., Crouseilles, N., Lemou, M., Méhats, F.: Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations. Numer. Math. 129(2), 211–250 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chartier, P., Méhats, F., Thalhammer, M., Zhang, Y.: Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations. Math. Comput. 85(302), 2863–2885 (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Cohen, D., Jahnke, T., Lorenz, K., Lubich, C.: Numerical integrators for highly oscillatory Hamiltonian systems: a review. In: Mielke, A. (ed.) Analysis, Modeling and Simulation of Multiscale Problems, pp. 553–576. Springer, Berlin (2006)Google Scholar
  13. 13.
    Eilinghoff, J., Schnaubelt, R., Schratz, K.: Fractional error estimates of splitting schemes for the nonlinear Schrödinger equation. J. Math. Anal. Appl. 442(2), 740–760 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Engquist, B., Fokas, A., Hairer, E., Iserles, A.: Highly Oscillatory Problems, 1st edn. Cambridge University Press, New York (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Faou, E.: Geometric Numerical Integration and Schrödinger Equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2012)CrossRefzbMATHGoogle Scholar
  16. 16.
    Faou, E., Gauckler, L., Lubich, C.: Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation. In: Forum of Mathematics, Sigma, vol. 2, p. 45 (2014)Google Scholar
  17. 17.
    Faou, E., Gradinaru, V., Lubich, C.: Computing Semiclassical quantum dynamics with Hagedorn wavepackets. SIAM J. Sci. Comput. 31(4), 3027–3041 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gabitov, I., Turitsyn, S.K.: Breathing solitons in optical fiber links. J. Exp. Theor. Phys. Lett. 63(10), 861–866 (1996)CrossRefGoogle Scholar
  19. 19.
    Gabitov, I.R., Turitsyn, S.K.: Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation. Opt. Lett. 21(5), 327–329 (1996)CrossRefGoogle Scholar
  20. 20.
    García-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20(3), 930–963 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gauckler, L.: Error analysis of trigonometric integrators for semilinear wave equations. SIAM J. Numer. Anal. 53(2), 1082–1106 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gauckler, L., Lubich, C.: Splitting integrators for nonlinear Schrödinger equations over long times. Found. Comput. Math. 10(3), 275–302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Green, W.R., Hundertmark, D.: Exponential decay of dispersion-managed solitons for general dispersion profiles. Lett. Math. Phys. 106(2), 221–249 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Grimm, V., Hochbruck, M.: Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A 39(19), 5495–5507 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  26. 26.
    Hansen, E., Ostermann, A.: High-order splitting schemes for semilinear evolution equations. BIT Numer. Math. 56(4), 1303–1316 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83(3), 403–426 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hundertmark, D., Lee, Y.R.: Decay estimates and smoothness for solutions of the dispersion managed non-linear Schrödinger equation. Commun. Math. Phys. 286(3), 851–873 (2009)CrossRefzbMATHGoogle Scholar
  30. 30.
    Jahnke, T.: Numerische Verfahren für fast adiabatische Quantendynamik. Ph.D. thesis, Uni Tübingen (2003)Google Scholar
  31. 31.
    Jahnke, T.: Long-time-step integrators for almost-adiabatic quantum dynamics. SIAM J. Sci. Comput. 25(6), 2145–2164 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Jahnke, T., Lubich, C.: Numerical integrators for quantum dynamics close to the adiabatic limit. Numer. Math. 94(2), 289–314 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lin, C., Kogelnik, H., Cohen, L.G.: Optical-pulse equalization of low-dispersion transmission in single-mode fibers in the 1.3–1.7-microm spectral region. Opt. Lett. 5(11), 476–8 (1980)CrossRefGoogle Scholar
  34. 34.
    Lorenz, K., Jahnke, T., Lubich, C.: Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition. BIT Numer. Math. 45(1), 91–115 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77(264), 2141–2153 (2008)CrossRefzbMATHGoogle Scholar
  36. 36.
    Medvedev, S.B., Turitsyn, S.K.: Hamiltonian averaging and integrability in nonlinear systems with periodically varying dispersion. J. Exp. Theor. Phys. Lett. 69(7), 499–504 (1999)CrossRefGoogle Scholar
  37. 37.
    Mikl, M.: Time-integration methods for a dispersion-managed nonlinear Schrödinger equation. Ph.D. thesis, Karlsruher Institute for Technology (2017)Google Scholar
  38. 38.
    Mollenauer, L.F., Grant, A., Liu, X., Wei, X., Xie, C., Kang, I.: Experimental test of dense wavelength-division multiplexing using novel, periodic-group-delay-complemented dispersion compensation and dispersion-managed solitons. Opt. Lett. 28(21), 2043–2045 (2003)CrossRefGoogle Scholar
  39. 39.
    Mollenauer, L.F., Mamyshev, P.V., Gripp, J., Neubelt, M.J., Mamysheva, N., Grüner-Nielsen, L., Veng, T.: Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons. Opt. Lett. 25(10), 704–706 (2000)CrossRefGoogle Scholar
  40. 40.
    Moloney, J.V., Newell, A.C.: Nonlinear Optics. Westview Press, Boulder (2004)zbMATHGoogle Scholar
  41. 41.
    Pelinovsky, D., Zharnitsky, V.: Averaging of dispersion-managed solitons: existence and stability. SIAM J. Appl. Math. 63(3), 745–776 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numer. 6, 437–483 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Runborg, O.: Mathematical models and numerical methods for high frequency waves. Commun. Comput. Phys. 2(5), 827–880 (2007)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Turitsyn, S.K., Bale, B.G., Fedoruk, M.P.: Dispersion-managed solitons in fibre systems and lasers. Phys. Rep. 521(4), 135–203 (2012)CrossRefGoogle Scholar
  46. 46.
    Turitsyn, S.K., Shapiro, E.G., Medvedev, S.B., Fedoruk, M.P., Mezentsev, V.K.: Physics and mathematics of dispersion-managed optical solitons. C. R. Phys. 4(1), 145–161 (2003)CrossRefGoogle Scholar
  47. 47.
    Werner, D.: Funktionalanalysis. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  48. 48.
    Wu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  49. 49.
    Zakharov, V.: Optical solitons: theoretical challenges and industrial perspectives. In: Les Houches Workshop, September 28–October 2, 1998. Springer EDP Sciences, Berlin New York Les Ulis, France Cambridge, MA (1999)Google Scholar
  50. 50.
    Zharnitsky, V., Grenier, E., Jones, C.K., Turitsyn, S.K.: Stabilizing effects of dispersion management. Physica D 152–153, 794–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute for Applied and Numerical MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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