Spectral Analysis und Applications to Solition Equations

From everyday experience we know that water waves show two types of phenomena: In regions where they can be described by linear equations, the so-called dispersion causes spreading of waves. In regions where nonlinear effects gain influence we see breaking of waves. All the more surprising was the observation of the young engineer John Scott Russell in 1834 that both effects can balance each other yielding waves which propagate without changing their shape. Such waves are known as solitons. It took until 1895 for this phenomenon to be explained theoretically by the Korteweg-de Vries equation (KdV). But only more than hundred years later their real importance was discovered: Around 1955 it was generally believed that the energy of a system of coupled oscillators would be dissipated equally among all eigenmodes by a small nonlinear perturbation. But much to everyone`s surprise the result of a computer experiment carried out by Enrico Fermi, John Pasta, and Stanislaw Ulam showed a quasi-periodic behavior of the energy distribution, that is, the energy distribution constantly returns almost to its initial value. It took another ten years until Martin Kruskal and Norman Zabuski laid the foundation for the explanation of this phenomenon by showing that the FPU experiment can be described by the KdV equation, that is, the very same equation which describes Russell`s water waves. Furthermore, they documented with further computer experiments that, independent of the initial wave shape, after some time only a number of solitons persist. In other words, solitons are the stable part of KdV solutions! The mathematical solution of the KdV equation was given shortly after by Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura with the help of the inverse scattering theory from quantum mechanics, thereby linking two previously unconnected fields. Peter Lax finally introduced a unified approach which allowed the extension to other soliton equations. Since then, this fascinating area has attracted enormous interest and a huge amount of literature and applications has accumulated. In optical fibers, for example, solitons are nowadays used to achieve transmission rates (depending on the distance) of up to 10TBit per second. Most articles assume a constant background (no excitation far outside). The case of solitons traveling on a periodic carrier wave is still in its infancy and involves many open questions. It is the main objective of this project to contribute to the solutions of these questions. The accompanying mathematical problems are of relevance to both quantum mechanics (scattering theory in crystals (metals) respectively two combined semi-infinite crystals) and nonlinear optics.

The aim of the project was to develop mathematical tools which help in understanding fundamental models in physics (in particular from quantum mechanics and wave phenomena) and biology.The first part of the project was concerned with spectral theory and scattering theory for one-dimensional quantum mechanical systems. In particular, we investigated the problem of an electron moving in a (quasi-)periodic structure (a simple model for a metal) as well as the case of two different semi-infinite (quasi-)periodic structures (a simple model for two different connected metals) and were able to develop a complete direct and inverse scattering theory for these situations. This includes the question on how to reconstruct the precise structure of the system from measured quantities like the transmission and reflection properties. We also developed novel tools for the investigation of eigenvalues near the edges of spectral bands in such models. Furthermore, we developed an approach for direct and inverse spectral problems where the potential has a strong singularity at a finite boundary point. This covers in particular the important case of radial operators, that is, the radial component of a three dimensional spherically symmetric problem.The second part of the project was concerned with completely integrable nonlinear wave equations. The problem here is that nonlinear equations that model wave phenomena (like for example water waves) cannot be solved explicitly and this makes them difficult to analyze (e.g., to predict the evolution of a wave for large times). On the other hand, many of these models are close to a completely integrable model, which has an additional mathematical structure which allows it to be solved explicitly (thus explaining the name). A detailed investigation of the structure of the solution of the integrable model is hence crucial for our understanding of real-world problems. During the project we obtained a complete description of all algebro-geometric solutions of the Toda and Ablowitz-Ladik equations (which led to a research monograph published with Cambridge University Press) and gave a detailed description of asymptotically periodic solutions of the Toda (the integrable model related to the famous Fermi-Pasta-Ulam problem) and the Korteweg-de Vries equation (the most famous soliton equation). We also gave a complete description of the long-time behavior of shock and rarefaction waves for these equations.The third part of the project dealt with mathematical modeling of the human respiratory system. In a nutshell this is a solid theoretical background to what is for example done during an alcohol test, namely the estimation of the blood alcohol concentration from a breath sample. The advance of different analytical methods in mass spectrometry within the last twenty years has opened the door to breath gas analysis and there is considerable evidence that volatile organic compounds (VOCs) produced in the human body and then partially released in breath have great potential for diagnosis in physiology and medicine. Breath gas is easily collectible and sampling is non-invasive in contrast to blood tests. It could be even done in real time allowing to monitor biological processes in the body. The aim of the project was to develop new mathematical models for the human respiratory system which allow to relate blood concentrations of certain marker substances to their corresponding breath concentrations. While this sounds fairly simple (and hence was often not taken into account properly in the past), it turns out to be a subtle issue for substances with high blood: air partition coefficients (which are hence not well described by the classical Farhi equation). During the project we were able to shed some light on the production of isoprene within the human body, study the possible use for anesthetic monitoring, or determine the potential of various VOCs for urban search-and-rescue operations.