NFT ALGORITHMS
Most existing algorithms were of second order, i.e. the numerical error depends quadratically on the sampling interval. New fast higher-order algorithms for NFTs of the nonlinear Schroedinger equation (NSE) and, respectively, the Korteweg-de Vries (KdV) equation were developed [Chimmalgi et al., IEEE Access, 2019], [Prins and Wahls, submitted]. The norming constants are a part of the nonlinear spectrum that is especially difficult to compute. A new method for their computation that finally succeeds in cases where prior work failed was presented in [Prins and Wahls, IEEE Access 2019]. Specialized (inverse) NFT algorithms that targeted at applications discussed below were presented in [Prins and Wahls, Commun Nonlin Sci Numer Simul 2021] (bore wave reconstruction); [Wahls, ECOC'17], [Chimmalgi and Wahls, ECOC'19] and [Wahls et al., OFC'19] (b-modulation); and [Chimmalgi and Wahls, in revision] and [Kitsios, TU Delft repository, 2022] (deep water rogue wave analysis). Many of the algorithms have been made publicly available in the open source software library FNFT [Wahls et al., J Open Source Softw 2018] and were used in the applications.
FIBER-OPTIC COMMUNICATION
Initially, there was no way to formulate finite duration constraints in the nonlinear Fourier domain, which led to inefficient pulse shapes. We proposed a new approach called b-modulation that allowed the generation of signals with a finite, pre-specified duration [Wahls, ECOC'17]. The maximum energy of such systems turned out to be finite. With collaborators, we investigated how this energy barrier could be shifted [Gui et al., Opt Expr 2018]. However, it still was not possible to generate long, high power signals, which are desirable for efficient communication. We discovered a theoretical explanation for this effect when we showed that power must go down when signal duration increases and nonlinear bandwidth is constant for the system above [Chimmalgi and Wahls, ECOC'19]. This new fundamental relation between duration, nonlinear bandwidth and power was generalized in [Chimmalgi and Wahls, Entropy 2020]. Since nonlinear spectra of efficient conventional communication signals are often dominated by solitons, a new method that turns a multi-soliton, which is always of infinite duration, into a finite duration signal by adding a suitable continuous spectrum was therefore developed [Wahls, ISWCS'21], [Wahls, OFC'22]. With contributors, we developed an open source simulation environment for fiber-optic transmissions [Brehler et al., OFC'19].
WATER WAVES
We studied the disintegration of initial bore waves into solitons with collaborators. We could demonstrate that the KdV-NFT predicts the characteristics of the later emerging solitons reliably over long distances [Bruehl et al., Wave Motion 2022]. Bores were recently shown experimentally to occur over fringing reefs that model e.g. coral reefs [Tissier et al., ICCE'18]. We are currently analyzing the data from these experiments with collaborators. We also investigated ship waves [Zhang et al., Coastal Dynamics'21], [Bruehl and Wahls, OMAE'22] and demonstrated that the NFT can be used to deduce the water depth from surface elevation measurements at two points [de Koster et al., accepted for OMAE'22]. Rogue waves are waves that are much higher than the surrounding waves. Together with collaborators, we have for the very first time systematically analyzed a large amount of real-world rogue waves using NFTs. In [Teutsch et al., NHESS Preprint 2022], 1172 rogue wave samples and 13.984 non-rogue wave samples measured at a site in the North Sea were systematically analyzed using the KdV-NFT. Certain characteristic nonlinear spectra were found to be strong predictors for large rogue waves. Together with other collaborators, we have also classified 663 real-world rogue waves that were measured at a site in the Philippine Sea using the NSE-NFT [Lee at al., in preparation]. The tutorial paper [Wahls, OMAE'20] finally provides the complete background needed to analyze deep water time series using our software library FNFT.
