Reliable computation of the eigenvalues of the discrete KdV spectrum

Peter J. Prins and Sander Wahls

Applied Mathematics and Computation, 2022, vol. 433, issue C

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.

Keywords: Korteweg–de Vries equation (KdV); Nonlinear Fourier Transform (NFT); Eigenvalue; Schrödinger equation; Sturm–Liouville equation; Sampled signal (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322004350

DOI: 10.1016/j.amc.2022.127361

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