In this paper, we study the propagation of bores over a long distance. We employ experimental data as input for numerical simulations using COULWAVE. The experimental flume is extended numerically to an effective relative length of x/h=3000, which allows all far-field solitons to emerge from the undular bore in the simulation data. We apply the periodic KdV-based nonlinear Fourier transform (KdV-NFT) to the time series taken at different numerical gauges and compare the results with those of the conventional Fourier transform. We find that the periodic KdV-NFT reliably predicts the number and the amplitudes of all far-field solitons from the near-field data long before the solitons start to emerge from the bore, even though the propagation is only approximated by the KdV. It is the first time that the predictions of the KdV-NFT are demonstrated over such long distances in a realistic set-up. In contrast, the conventional linear FT is unable to reveal the hidden solitons in the bore. We repeat our analyses using space instead of time series to investigate whether the space or time version of the KdV provides better predictions. Finally, we show how stepwise superposition of the determined solitons, including the nonlinear interactions between individual solitons, returns the analysed initial bore data.@en

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.@en

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.@en

We present an algorithm to compute the N-fold Crum transform (also known as the dressing method) for the Korteweg–de Vries equation (KdV) accurately in floating point arithmetic. This transform can be used to generate solutions of the KdV equation, e.g. as a part of the inverse Non-linear Fourier Transform. Crum transform algorithms that sequentially add the N eigenvalues to the solution with a chain of N Darboux transforms have a computational complexity of O(N ²), but suffer inevitably from singular intermediate results during the computation of certain regular Crum transforms. Algorithms that add all N eigenvalues at once do not have that flaw, but have a complexity of O(N ³) and are often even less accurate for other reasons. Our algorithm has a complexity of O(N ²). It makes use of a chain of 2-fold Crum transforms and, if N is odd, one Darboux transform. Hence, our algorithm adds two eigenvalues at a time instead of one whenever possible. We prove that with the right eigenvalue ordering, this avoids artificial singularities for all regular Crum transforms. Furthermore, we demonstrate that our algorithm is considerably more accurate in floating point arithmetic than benchmark algorithms found in the literature. At the same error tolerance, N can be three to seven times as high when using our algorithm instead of the best among the benchmark algorithms.

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Several non-linear fluid mechanical processes, such as wave propagation in shallow water, are known to generate solitons: localized waves of translation. Solitons are often hidden in a wave packet at the beginning and only reveal themselves in the far-field. With a special signal processing technique known as the non-linear Fourier transform (NFT), solitons can be detected and characterized before they emerge. In this paper, we present a new algorithm aimed at computing the phase shift of solitons in processes governed by the Korteweg–de Vries (KdV) equation. In numerical examples, the new algorithm is found to perform reliably even in cases where existing algorithms break down.@en

The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of highly accurate low-complexity algorithms remains a challenge. In this paper, we present new fast forward NFT algorithms that achieve accuracies that are orders of magnitudes better than current methods, at comparable run times and even for moderate sampling intervals. The new algorithms are compared to existing solutions in multiple, extensive numerical examples.@en

A numerical method for the generation of fiber inputs in nonlinear frequency division multiplexing (NFDM) systems based on b-modulation is provided. The method is parallelizable, does not suffer from error propagation, and converges exponentially.

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We present a nonlinear Fourier transform algorithm whose accuracy, at a comparable runtime and for moderate step sizes, is orders of magnitude better than that of the classical Boffetta-Osborne method.@en

Non-linear Fourier Transforms (NFTs) enable the analysis of signals governed by certain non-linear evolution equations in a way that is analogous to how the conventional Fourier transform is used to analyse linear wave equations. Recently, fast numerical algorithms have been derived for the numerical computation of certain NFTs. In this paper, we are primarily concerned with fast NFTs with respect to the Korteweg-de Vries equation (KdV), which describes e.g. the evolution of waves in shallow water. We find that in the KdV case, the fast NFT can be more sensitive to numerical errors caused by an exponential splitting. We present higher order splittings that reduce these errors and are compatible with the fast NFT. Furthermore we demonstrate for the NSE case that using these splittings can make the accuracy of the fast NFT match that of the conventional NFT.@en

The conventional Fourier transform was originally developed in order to solve the heat equation, which is a standard example for a linear evolution equation. Nonlinear Fourier transforms (NFTs)1 are generalizations of the conventional Fourier transform that can be used to solve certain nonlinear evolution equations in a similar way (Ablowitz et al. 1974). An important difference to the conventional Fourier transform is that NFTs are equationspecific. The Korteweg-de Vries (KdV) equation (Gardner et al. 1967) and the nonlinear Schroedinger equation (NSE) (Shabat and Zakharov 1972) are two popular examples for nonlinear evolution equations that can be solved using appropriate NFTs.@en