  # Nonlinear Fourier Transforms in Action

## Periodic Reporting for period 3 - NEUTRINO (Nonlinear Fourier Transforms in Action)

Reporting period: 2020-03-01 to 2021-08-31

Linearity is of the most prolific concepts in engineering. In a nutshell, it means that the various quantities describing a system can be expressed as weighted sums of one another. The change in the level of a water tank can for example be modeled as the difference between the amount of inflowing water and the amount of outflowing water. 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. Solitons 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, optical fibers and the Earth's magentosphere. Nonlinearity is essential for the formation of solitons in all of these examples. It was hence a sensation when researchers started to learn that many of these inherently nonlinear systems have deeply hidden underlying linearities. Instead of considering the physical spatio-temporal nonlinear system directly, one can instead consider two separate linear systems, one spatial and one temporal, that are coupled by a nonlinear compatibility condition. By spectral analysis of these linear systems, we can decompose signals governed by the original nonlinear system into physically meaningful components that evolve in simple ways. Since this type of spectral analysis generalizes the conventional Fourier transform, a fundamental tool in linear systems theory, it is also called "nonlinear Fourier transform" in the literature.

Nonlinear Fourier transforms have so far mostly been considered from theoretical viewpoints. The goal of the ERC Starting Grant NEUTRINO (NonlinEar FoUrier TRansforms IN actiOn) is to bridge the gap to applications and establish nonlinear Fourier transforms as a practical tool in engineering. Even though a plethora of practically relevant nonlinear systems could in principle be approached with nonlinear Fourier transforms, this currently happens only rarely in practice. Interested engineers first have to acquire the necessary theoretical background and then implement numerical algorithms before they can even start applying nonlinear Fourier transforms to practical problems. These are time-consuming tasks. Naive numerical implementations are furthermore slow and suffer typically from large execution times. All of this significantly hinders the broader adoption of nonlinear Fourier transforms in engineering. In NEUTRINO, we hence develop fast algorithms for the numerical computation of nonlinear Fourier transforms and showcase them while working on two prototypical engineering problems that are promising for using nonlinear Fourier transforms: fiber-optic communications and water wave data analysis. The developed algorithms are made available to the public as open source software.
When the project started, several fiber-optic communication systems based on nonlinear Fourier transforms had already been proposed and investigated. In these systems, the transmitter starts by embedding the user's information in a nonlinear Fourier spectrum. It then computes the inverse nonlinear Fourier transform of that spectrum and injects the resulting time domain signal into the nonlinear fiber-optic link. The receiver at the other end of the link computes the forward nonlinear Fourier transform of the received signal and extracts the information. The last step is simple because reverting the impact of dispersion and the nonlinear Kerr effect in the fiber is easy in the nonlinear Fourier domain.

A fundamental problem at the time was that in practical systems, both the transmitted and received signal should fit into the finite processing windows at the transmitter and receiver. There was however no way to formulate this constraint in the nonlinear Fourier domain. The nonlinear Fourier spectra generated at the receiver corresponded to signals of infinite duration, which thus had to be truncated. The truncation error was difficult to control and often significant. The first major contribution within the first half of this project was a new way to embed data in a nonlinear Fourier spectrum, called b-modulation, that allowed transmitters to generate signals of a finite, pre-specified duration. Solving a major problem, b-modulation was soon picked up by other groups and is now a well-known method in the field.

We continued investigating b-modulation and discovered a peculiar new effect that we called the "energy barrier": the maximum energy that a b-modulation system can produce depends on the carrier waveform. Working with Prof. A.P.T. Lau's group from The Hong Kong Polytechnic University, we investigated how the energy barrier could be shifted. The resulting system was investigated both in simulations and experimentally. However, even shifting the energy barrier did not resolve another so far somewhat mysterious effect that had been observed in several scenarios. The overhead due to the gaps that are needed to separate data carrying bursts should be negligible, which means that burst durations should be long. However, when long signals were generated in practice, their power turned out to be too low to support an efficient communication. While this effect had already been observed in simulations for several system designs, the underlying reasons were not clear. We provided the first theoretically sound explanation when we showed that for the b-modulation system mentioned above, it is a unavoidable consequence of the energy barrier.

The open source software library FNFT is another major result of the project. FNFT contains implementations of numerical algorithms to compute nonlinear Fourier transforms. It can be obtained at https://github.com/FastNFT/FNFT/ . Already before the project started, we had developed several fast algorithms to compute nonlinear Fourier transforms numerically, similar to how the famous fast Fourier transform (FFT) algorithm speeds up the computation of the conventional Fourier transform. The first release of FNFT contained implementations these algorithms. To be best of our knowledge, it was the first implementation available to the public. The next release added some fast inverse nonlinear Fourier transforms, among them a fast algorithm needed to perform b-modulation in fiber-optic communications. Again, this was the first time such algorithms became available to the public, enabling researchers around the world to start investigating nonlinear Fourier transforms with minimal effort.

Together with two external contributors to FNFT, M. Brehler and C. Mahnke, we also developed an open source simulation environment for fiber-optic transmissions based on nonlinear Fourier transforms called NFDMLab. The two goals of NFDMLab were to 1) showcase the results achieved in the project so far (it is based on FNFT), and 2) serve as a foundation for upcoming research in our and potentially other groups. NFDMLab has been presented to the fiber-optics community in the Demo Zone of the 2019 Optical Fiber Communication Conference and Exhibition (OFC), which is the largest event in the field.

Finally, a new highly efficient fast nonlinear Fourier transform algorithm that requires significantly fewer samples to reach low errors has been developed. We are currently working on integrating it into FNFT.

In the following, we will discuss results related to water wave data analysis. At this point, we remark that there is more than one nonlinear Fourier transform. The right choice of transform depends on the underlying physics. The nonlinear Fourier transforms used in the works on fiber-optic communications rely on the assumption that the underlying physics can be described using the nonlinear Schroedinger equation. We now consider waves in shallow water that can be modeled by the Korteweg-de Vries equation.

We are also working on developing fast algorithms for the nonlinear Fourier transform with respect to the Korteweg-de Vries equation. While this case is in many regards similar to the nonlinear Schroedinger case considered in fiber-optic communications, there are also non-trivial differences. Many different mathematical formulations of the transform are known in the literature, but not all of them are equally well-suited for numerical tasks. We have investigated the different possible approaches and worked out their advantages and disadvantages. An important discovery was made when the problem of computing soliton phase shifts was considered, which is essential for predicting the future position of currently hidden solitons using the current nonlinear Fourier spectrum. It was found that current approaches to this problem fail in certain scenarios. A new algorithm was devised that does not suffer from this problem.
The proposed b-modulation method extended the state of the art as it solved an so far open fundamental problem in fiber-optic communication (generation of finite duration signals) in a simple way. It has been picked up by many other groups. The energy barrier and transmit power limitations we uncovered were described for the first time in the literature. Hereby, we finally made progress on the theoretical understanding of another practically important problem (long burst durations lead to bad performance). In the second half of the project, we will investigate potential ways to the resolve these issues.

Several new nonlinear Fourier transform algorithms have been developed that were either significantly more efficient or reliable than existing algorithms. We are working on integrating them in FNFT, which is the first publicly available software library for the fast numerical computation of nonlinear Fourier transforms. Similarly, NFDMLab is the only publicly available simulation environment for nonlinear Fourier transform-based fiber-optic communications. We will continue to investigate new algorithms and to extend both FNFT and NFDMLab.

We will start to analyze water wave related data with the developed algorithms for the Korteweg-de Vries equation.