Generalized Indefinite Strings and Their Applications

The project concerns nonlinear wave equations, completely integrable systems and inverse spectral theory. More specifically, the main objects of our project are: (a) The CamassaHolm equation, one of the most important 1 + 1 dimensional completely integrable nonlinear wave equations. Among many celebrated models for shallow water waves (for example, the Kortewegde Vries equation or the BenjaminBonaMahoney equation), its significance stems from the fact that it is the first and long sought-for model with the following three features: complete integrability, as it admits a Lax pair formulation, solitons, especially including peaked ones (called peakons), and finite time blow-up of smooth solutions that resembles wave- breaking to some extent. Our overall goal aims in understanding conservative solutions of the CamassaHolm equation, a special type of global weak solutions preserving energy, by means of the inverse scattering/spectral transform. Notice that this task includes several long standing open problems in the area. (b) Generalized indefinite strings. This object was introduced in 2014 in our joint work with J. Eckhardt in connection with the isospectral problem for the conservative CamassaHolm flow. The importance of the latter stems from the fact that it serves as a canonical model of a self-adjoint operator with simple spectrum (two other classical models are Jacobi matrices and canonical systems). Generalized indefinite strings is our major tool. Notice that this novel object in direct and inverse spectral theory has several advantages compared to the classical spectral theory objects, like one dimensional Schrödinger equation (actually, the latter can be included into the theory of the former as a special case).