START Project: Spectral Analysis and Applications to Soliton 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.

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