The inverse spectral transform for the Camassa-Holm equation

Ever since its discovery by Clifford Gardner, John Greene, Martin Kruskal and Robert Miura in 1967, the Inverse Scattering/Spectral Transform has been developed into a powerful tool for solving completely integrable systems. The essence of this method lies in the observation that the flow of particular non-linear wave equations can be transformed into a simple linear flow of certain scattering or spectral data for an associated family of differential operators. Initially introduced in order to solve the Kortewegde Vries equation, this method has successfully been generalized to several other non-linear wave equations by now. One of these equations of current interest is the CamassaHolm equation, which models unidirectional wave propagation on shallow water above a flat bottom. The most intriguing property of the CamassaHolm equation is the fact that it also incorporates the phenomenon of wave breaking. This means that even smooth initial data may lead to blow-up in finite time, which happens in such a way that the solution stays bounded but its slope becomes vertical. Due to these remarkable properties, a lot of work has been devoted to the CamassaHolm equation during the last two decades. In particular, it has been shown that it is always possible to continue solutions beyond wave breaking upon considering suitable weak solutions. However, in order to guarantee uniqueness, one (for example) has to require that the energy of solutions is conserved, which leads to the notion of global conservative solutions of the CamassaHolm equation. The corresponding scattering/spectral problem for the CamassaHolm equation is a particular SturmLiouville problem with an (in general) indefinite weight. A lot of difficulties arise due to this indefiniteness, which is the main reason why results in this case are still quite scarce. The main objective of this project is to further develop the Inverse Scattering/Spectral Transform for the CamassaHolm equation. More precisely, I propose to study a generalized scattering/spectral problem, which is supposed to give rise to an Inverse Scattering/Spectral Transform for global conservative solutions of the CamassaHolm equation. This newly suggested approach seems very promising and I expect it to point the way ahead to future developments with respect to the conservative Camassa Holm equation. The main tasks include investigating the direct and inverse problems corresponding to this generalized scattering/spectral problem as well as relating it to the conservative CamassaHolm flow.

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 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.