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Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic, inverse scattering transform

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  • Christov, Ivan C.

Abstract

Recent numerical work on the Zabusky–Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne’s nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the exact number of solitons, their amplitudes and their reference level is computed. Other “less nonlinear” oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.

Suggested Citation

  • Christov, Ivan C., 2012. "Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic, inverse scattering transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1069-1078.
    • Handle: RePEc:eee:matcom:v:82:y:2012:i:6:p:1069-1078
    DOI: 10.1016/j.matcom.2010.05.021
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    References listed on IDEAS

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    1. Ilison, Lauri & Salupere, Andrus, 2009. "Propagation of sech2-type solitary waves in hierarchical KdV-type systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(11), pages 3314-3327.
    2. Zabusky, N.J. & Silver, D. & Fernandez, V., 1996. "Visiometrics and modeling in computational fluid dynamics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 40(3), pages 181-191.
    3. Salupere, A. & Engelbrecht, J. & Ilison, O. & Ilison, L., 2005. "On solitons in microstructured solids and granular materials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(5), pages 502-513.
    4. Salupere, A. & Engelbrecht, J. & Peterson, P., 2003. "On the long-time behaviour of soliton ensembles," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 137-147.
    5. Osborne, A.R., 1994. "Automatic algorithm for the numerical inverse scattering transform of the Korteweg–de Vries equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 37(4), pages 431-450.
    6. Christov, Ivan, 2009. "Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(1), pages 192-201.
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