Journal of Engineering Mathematics

, Volume 87, Issue 1, pp 167–186 | Cite as

Detailed comparison of numerical methods for the perturbed sine-Gordon equation with impulsive forcing

  • Danhua Wang
  • Jae-Hun Jung
  • Gino Biondini


The properties of various numerical methods for the study of the perturbed sine-Gordon (sG) equation with impulsive forcing are investigated. In particular, finite difference and pseudo-spectral methods for discretizing the equation are considered. Different methods of discretizing the Dirac delta are discussed. Various combinations of these methods are then used to model the soliton–defect interaction. A comprehensive study of convergence of all these combinations is presented. Detailed explanations are provided of various numerical issues that should be carefully considered when the sG equation with impulsive forcing is solved numerically. The properties of each method depend heavily on the specific representation chosen for the Dirac delta—and vice versa. Useful comparisons are provided that can be used for the design of the numerical scheme to study the singularly perturbed sG equation. Some interesting results are found. For example, the Gaussian approximation yields the worst results, while the domain decomposition method yields the best results, for both finite difference and spectral methods. These findings are corroborated by extensive numerical simulations.


Finite difference methods Impulsive forcing Sine-Gordon equation Spectral methods 



We thank Roy Goodman, Panayotis Kevrekidis, and Jingbo Xia for many valuable discussions. This study was partially supported by the National Science Foundation under Award Number DMS-0908399.


  1. 1.
    Scott AC, Chu FYF, McLaughlin DW (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–1483ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  3. 3.
    Fei Z (1992) Y S Kivshar and L Vázquez, Resonant kink–impurity interaction in the sine-Gordon model. Phys Rev A 45:6019–6030ADSCrossRefGoogle Scholar
  4. 4.
    Forinash K, Peyrard M, Malomed BA (1994) Interaction of discrete breathers with impurity modes. Phys Rev E 49:3400–3411ADSCrossRefGoogle Scholar
  5. 5.
    Cao XD, Malomed BA (1995) Soliton-defect collisions in the nonlinear Schrödinger equation. Phys Lett A 206:177–182ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Ernst T, Brand J (2010) Resonant trapping in the transport of a matter-wave soliton through a quantum well. Phys Rev A 81(033614):1–11Google Scholar
  7. 7.
    Goodman RH, Holmes PJ, Weinstein MI (2002) Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model. Physica D 161:21–44ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Goodman RH, Slusher RE, Weinstein MI (2002) Stopping light on a defect. J Opt Soc Am B 19:1635–1652ADSCrossRefGoogle Scholar
  9. 9.
    Goodman RH, Haberman R (2004) Interaction of sine-Gordon kinks with defects: the two-bounce resonance. Physica D 195:303–323ADSCrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Goodman RH, Haberman R (2007) Chaotic scattering and the \(n\)-bounce resonance in solitary–wave interactions. Phys Rev Lett 98(104103):1–4Google Scholar
  11. 11.
    Fratalocchi A, Assanto G (2007) Symmetry-breaking instabilities in perturbed optical lattices: nonlinear nonreciprocity and macroscopic self-trapping. Phys Rev A 75(063828):1–5MathSciNetGoogle Scholar
  12. 12.
    Mak WCK, Malomed BA, Chu PL (2003) Interaction of a soliton with a local defect in a Bragg grating. J Opt Soc Am B 20:725–735ADSCrossRefGoogle Scholar
  13. 13.
    Morales-Molina L, Vicencio RA (2006) Trapping of discrete solitons by defects in nonlinear waveguides arrays. Opt Lett 31:966–968ADSCrossRefGoogle Scholar
  14. 14.
    Pando CL, Doedel EJ (2005) Onset of chaotic symbolic synchronization between population inversions in an array of weakly coupled Bose–Einstein condensates. Phys Rev E 71(5 Pt 2): 056201Google Scholar
  15. 15.
    Peyrard M, Kruskal MD (1984) Kink dynamics in the highly discrete sine-Gordon system. Physica D 14:88–102ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    Piette B, Zakrzewski WJ (2007) Scattering of sine-Gordon kinks on potential wells. J Phys A 40(5995):1–16Google Scholar
  17. 17.
    Soffer A, Weinstein MI (2005) Theory of nonlinear dispersive waves and selection of the ground state. Phys Rev Lett 95(213905):1–4Google Scholar
  18. 18.
    Trombettoni A (2003) Discrete nonlinear Schrödinger equation with defects. Phys Rev E 67(016607):1–11MathSciNetGoogle Scholar
  19. 19.
    Friedlander FG, Joshi MS (1998) Introduction to the theory of distributions. Cambridge University Press, CambridgeGoogle Scholar
  20. 20.
    Ablowitz MJ, Herbst BM, Schober CM (1995) Numerical simulation of quasi-periodic solutions of the sine-Gordon equation. Phycica D 87:37–47ADSCrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Jung J-H, Don WS (2009) Collocation methods for hyperbolic partial differential equations with singular sources. Adv Appl Math Mech 1:769–780MathSciNetGoogle Scholar
  22. 22.
    Boyd JP (2001) Chebyshev and Fourier spectral methods. Dover, New YorkMATHGoogle Scholar
  23. 23.
    B Fornberg (1998) A practical guide to pseudospectral methods. Cambridge Universiy Press, CambridgeGoogle Scholar
  24. 24.
    Gottlieb D (1977) Numerical analysis of spectral methods: theory and applications. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  25. 25.
    Hesthaven JS, Gottlieb S, Gottlieb D (2007) Spectral methods for time-dependent problems. Cambridge Universiy Press, CambridgeCrossRefMATHGoogle Scholar
  26. 26.
    Trefethen LN (2000) Spectral methods in Matlab. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  27. 27.
    Chakraborty D, Jung J-H (2013) Efficient determination of the critical parameters and the statistical quantities for Klein–Gordon and sine-Gordon equations with a singular potential using generalized polynomial chaos methods. J Comput Sci 4:46–61CrossRefGoogle Scholar
  28. 28.
    Chakraborty D, Jung J-H, Lorin E (2013) Efficient determination of critical parameters of nonlinear Schrödinger equation with point-like potential using generalized polynomial chaos methods. App Numer Math 72:115–130CrossRefMathSciNetGoogle Scholar
  29. 29.
    Faddeev LD, Takhtajan LA (1987) Hamiltonian methods in the theory of solitons. Springer, New YorkCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsState University of New York at BuffaloBuffaloUSA

Personalised recommendations