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# Nonlinear Fourier Transforms in Action

## Periodic Reporting for period 4 - NEUTRINO (Nonlinear Fourier Transforms in Action)

Reporting period: 2021-09-01 to 2022-02-28

Linearity is one of the most prolific concepts in engineering. Linear systems are very well understood and can be solved efficiently both analytically and numerically. Nonlinear systems can often be linearized, that is, approximated by linear ones. However, there are limits to what linearization can describe. Engineering problems that cannot be linearized are often very difficult to work with.

An interesting example for a seemingly intrinsically nonlinear phenomenon are solitons. They are particle-like traveling waves that occur only when a nonlinear effect cancels linear dispersive effects. They occur in many different media including water canals and optical fibers. Nonlinearity is essential for the formation of solitons, but in many cases there is a hidden underlying linearity. Instead of considering the nonlinear system directly, one instead considers two separate linear systems that are coupled by a nonlinear compatibility condition. By spectral analysis of these linear systems, we can decompose signals governed by the nonlinear system into physically meaningful components that evolve in simple ways. This type of spectral analysis generalizes the conventional Fourier transform, and is therefore also called nonlinear Fourier transform (NFT).

The conventional Fourier transform is ubiquitous in engineering, while nonlinear Fourier transforms so far have mostly been considered from theoretical viewpoints. The goal of the project was to bridge the gap to applications and establish nonlinear Fourier transforms as a practical tool in engineering. We therefore developed efficient numerical methods for the numerical computation of nonlinear Fourier transforms, and showcased them in two exemplary engineering fields: fiber-optic communications and water wave data analysis.
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.
- The fast higher-order NFT algorithms were the first that showed improvements over conventional methods in the practically relevant moderate sampling interval regime.

- The algorithm for the norming constants is the first that is reliable in all investigated cases.

- FNFT is the first publicly available software library for the numerical computation of NFTs.

- The b-modulation method solved an so far open fundamental problem in fiber-optic communication in a simple way. It has been picked up by many other groups.

- The energy barrier and transmit power limitations for b-modulation were completely new phenomena that have no analogues in linear systems.

- The (multi-)soliton shortening method is another completely new approach for embedding data in the nonlinear spectra of finite duration pulses.

- For first time, data sets with large numbers of real-world rogue wave data were analyzed using NFTs. The reviews of the preprint are public. One reviewer noted the following: "Clearly, the topic is of the highest relevance and of cutting-edge nature in ocean sciences, while also being within the scope of the journal. Provided major amendments are implemented, I believe the revised version of this paper would be an essential reading for everyone studying extreme waves in the ocean."

The project has ended, but there is still ongoing activity. This is mostly related to manuscripts that still need to be submitted or revised. An up to date overview can be found on the project homepage.