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Reliable computation of the eigenvalues of the discrete KdV spectrum

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  • Prins, Peter J.
  • Wahls, Sander

Abstract

We propose a numerical algorithm that computes the eigenvalues of the Korteweg–de Vries equation (KdV) from sampled input data with vanishing boundary conditions. It can be used as part of the Non-linear Fourier Transform (NFT) for the KdV equation. The algorithm that we propose makes use of Sturm Liouville (SL) oscillation theory to guaranty that all eigenvalues are found. In comparison to similar available algorithms, we show that our algorithm is more robust to numerical errors and thus more reliable. Furthermore we show that our root finding algorithm, which is based on the Newton–Raphson (NR) algorithm, typically saves computation time compared to the conventional approaches that rely heavily on bisection.

Suggested Citation

  • Prins, Peter J. & Wahls, Sander, 2022. "Reliable computation of the eigenvalues of the discrete KdV spectrum," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    • Handle: RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322004350
    DOI: 10.1016/j.amc.2022.127361
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    References listed on IDEAS

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    1. Srikanth Sugavanam & Morteza Kamalian Kopae & Junsong Peng & Jaroslaw E. Prilepsky & Sergei K. Turitsyn, 2019. "Analysis of laser radiation using the Nonlinear Fourier transform," Nature Communications, Nature, vol. 10(1), pages 1-10, December.
    2. Vasily V. Temnov & Christoph Klieber & Keith A. Nelson & Tim Thomay & Vanessa Knittel & Alfred Leitenstorfer & Denys Makarov & Manfred Albrecht & Rudolf Bratschitsch, 2013. "Femtosecond nonlinear ultrasonics in gold probed with ultrashort surface plasmons," Nature Communications, Nature, vol. 4(1), pages 1-6, June.
    3. 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.
    4. 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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