The conservative Camassa-Holm flow

Soliton equations are typically nonlinear partial differential equations that arise in various areas of physics. From nonlinear optics to atomic physics and fluid mechanics, these equations are often known for their profound mathematical structures. The most prominent example of a soliton equation is arguably the intensively studied Korteweg-de Vries (KdV) equation. Derived in 1895 as a model for shallow water waves, the KdV equation allowed to explain previously observed occurrence of solitary waves as a subtle balance between nonlinear and dispersive effects. Around seventy years later, the theoretical framework for this phenomenon was found by Clifford Gardner, John Greene, Martin Kruskal and Robert Miura, who introduced the Inverse Scattering Method to solve the KdV equation, showing that the KdV equation may be viewed as an infinite dimensional completely integrable Hamiltonian system. This seminal discovery marked the beginning of what is nowadays known as soliton theory; an ever growing fascinating field of active research with rich and fruitful interactions from applied sciences. One particular soliton equation that has been studied extensively during the last two decades is the so-called Camassa-Holm (CH) equation. Its main relevance stems from the fact that it can be derived as a model for unidirectional wave propagation on shallow water. The most intriguing property of the CH equation is that, in contrast to the KdV equation, it allows for smooth initial data to blow up in finite time in a way that resembles wave breaking. Despite a lot of activity and a still growing enormous amount of literature on this equation, it is still not feasible to employ the Inverse Scattering Method for the CH equation. The proposed research will take advantage of recent advances in the theory of indefinite spectral problems (more precisely, indefinite generalized strings), pioneered by Mark Krein and Heinz Langer in the 1970s, to overcome the occurring complications. It is the main objective of this project to finally establish the Inverse Scattering Method for the CH equation as well as for related equations. This will subsequently lead to a new understanding of these equations as infinite dimensional completely integrable Hamiltonian systems.

Ever since its discovery, the Inverse Scattering/Spectral Transform has been developed into a powerful method for solving completely integrable systems. The essence of this approach lies in the observation that the flow of particular nonlinear wave equations can be transformed into a simple linear flow of certain scattering or spectral data for an associated family of differential operators. It was the principal objective of the project to implement this method for the Camassa-Holm equation, which arises as a model for unidirectional wave propagation on shallow water above a flat bottom. In particular, this involved to investigate direct and inverse spectral theory for a generalized Sturm-Liouville problem with an indefinite weight of low regularity. One of the most intriguing features of the Camassa-Holm equation is that it allows solutions to terminate after finite time in a way that resembles wave breaking to some extent. The complications encountered due to this blow-up are reflected by difficulties with the associated inverse spectral problem, which is the main reason why the Inverse Scattering/Spectral Transform has not been implemented before for the Camassa-Holm equation. During the course of the project, new methods for generalized Sturm-Liouville problems with indefinite weights of low regularity have been developed in order to overcome these problems. As a consequence, it was finally possible to establish the Inverse Spectral Transform for the Camassa-Holm equation with decaying initial data. The results obtained from this project led to a new understanding of the conservative Camassa-Holm flow as a completely integrable system.