Reliable numerical algorithms for the Non-linear Fourier Transform of the KdV equation

Reliable numerical algorithms for the Non-linear Fourier Transform of the KdV equation

Prins, Peter J. (TU Delft Team Sander Wahls)

Wahls, S. (promotor)
Verhaegen, M.H.G. (promotor)

Delft University of Technology

2022-04-28

Research question
The topic of this dissertation is the numerical computation of the forward and inverse Non-linear Fourier Transform (NFT) for the Korteweg–de Vries equation (KdV), for sampled signals that decay sufficiently fast on both sides. With NFTs certain non-linear Partial Differential Equations (PDEs) can be solved in a way that is analogous to solving linear Ordinary Differential Equations (ODEs) and PDEs by means of the ordinary Fourier transform. Similarly to the linear Fourier transform, NFTs can be used to analyse, synthesise, filter and predict signals. Existing numerical NFT algorithms suffer from either or both a limited accuracy or a long computation time, which limit the usability of the KdV-NFT for engineering problems. In this dissertation we develop new algorithms that achieve a higher accuracy or require a shorter computation time.

Design methods
We implemented existing numerical algorithms in Mathworks Matlab in floating point arithmetic to analyse their behaviour. Thereafter we designed new algorithms that avoid the undesirable behaviour of the existing algorithms. We demonstrated the improvements by means of benchmark tests. Furthermore we implemented some of the new algorithms in the programming language C in the Fast Non-linear Fourier Transform (FNFT) software library.
Results
We have developed algorithms to compute the continuous KdV-NFT spectrum and the eigenvalues and norming constants of the discrete KdV-NFT spectrum. Furthermore we developed an algorithm to compute the contribution of the discrete spectrum to the inverse KdV-NFT. The continuous KdV-NFT spectrum can now be computed with a fast algorithm at a comparable error tolerance to the Non-linear Schrödinger Equation (NSE)-NFT. That means that the computational complexity has been reduced from O(D^2) to O(D(log(D))^2), where D is the number of samples, without a significant deterioration of the accuracy. The eigenvalues of the discrete KdV-NFT spectrum can now be computed reliably and more efficiently than before. The norming constants can now be computed in all known cases without the anomalous errors that were observed for older algorithms. That means an improvement of the accuracy by several orders of magnitude. The contribution of the inverse KdV-NFT can now be computed for discrete spectra with three to seven times as many eigenvalues in comparison to previously available algorithms.
Conclusions and applications
The KdV can be used as a model for nearly linear wave phenomena that propagate in one direction. These are found in a plethora of physical applications. The algorithms that we presented in this dissertation can be used for the analysis, synthesis, filtering and prediction of sampled data from such systems. Their higher accuracy and/or shorter computation time thus brings the KdV-NFT a step closer to the engineering practice.

signal processing algorithms
non-linear Fourier transform (NFT)
Korteweg–de Vries (KdV) equation
Schrödinger equation
water wave
soliton
norming constant
exponential splittings
dressing method
Darboux transform
Crum transform

https://doi.org/10.4233/uuid:171ba94a-e8f4-4969-b6ed-912d4f334968

978-94-6384-320-1

Institutional Repository

doctoral thesis

© 2022 Peter J. Prins