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Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

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  • Christov, Ivan

Abstract

The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg–de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the “spectral eigenvalues” (i.e., the ±1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.

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  • 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.
    • Handle: RePEc:eee:matcom:v:80:y:2009:i:1:p:192-201
    DOI: 10.1016/j.matcom.2009.06.005
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Chin-Bing, S.A. & Warn-Varnas, A. & King, D.B. & Lamb, K.G. & Teixeira, M. & Hawkins, J.A., 2003. "Analysis of coupled oceanographic and acoustic soliton simulations in the Yellow Sea: a search for soliton-induced resonances," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 11-20.
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    Cited by:

    1. 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.
    2. Prins, Peter J. & Wahls, Sander, 2022. "Reliable computation of the eigenvalues of the discrete KdV spectrum," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    3. Salupere, Andrus, 2016. "On hidden solitons in KdV related systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 127(C), pages 252-262.

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