The Camassa-Holm equation and indefinite spectral problems

The history of solitary water waves dates back to the experimental work of Russell in 1844. The Korteweg-de Vries equation (KdV) was introduced in 1895 to model the behavior of long waves on shallow water. Shortly after the discovery of solitons by Zabusky and Kruskal in 1965, Gardner, Greene, Kruskal and Miura pioneered a new method for solving the KdV equation by invoking direct and inverse scattering theory from quantum mechanics. Subsequently Lax considerably generalized these ideas, and Zakharov and Shabat showed that the method worked for another physically significant nonlinear evolution equation, namely, the nonlinear Schrodinger equation. In 1993, Camassa and Holm derived a new equation and showed that, while on the one hand side it still exhibits soliton solutions and is solvable via the inverse scattering method, it is also capable of modeling wave breaking (a phenomenon not known for the KdV equation). In particular, the wave breaking may occur if the associated isospectral problem is indefinite. In this respect, the theory of direct and inverse scattering for the corresponding Sturm-Liouville problem is of crucial importance for the Camassa-Holm (CH) equation. Unfortunately, no such theory is available if the problem is indefinite. Moreover, Beals, Sattinger and Szmigielski have shown that in certain cases the inverse spectral problem is not solvable and this corresponds to the blow up. Motivated by our recent study of multi-peakon solutions, we suggest a new generalized spectral problem, which is quadratic in a spectral parameter, as an isospectral problem for the conservative CH equation. The aims of the project are to develop direct and inverse scattering theory for this generalized indefinite spectral problem and to study the blow up phenomena for the CH equation with the help of the inverse scattering transform.

The Camassa-Holm (CH) equation is a nonlinear PDE arising in the shallow water theory. This equation is formally integrable in the sense that it admits the Lax pair structure and the corresponding isospectral problem is the so-called inhomogeneous Krein string, a weighted Sturm-Liouville problem. In contrast to the Korteweg-de Vries (KdV) equation, the CH equation possesses breaking waves and this happens precisely when the associated spectral problem is indefinite. The overall aim of the project is to understand the intriguing features of solutions to the CH equation via the inverse scattering transform approach. The key ingredient is the direct and inverse spectral theory of the corresponding isospectral problem. Jointly with J. Eckhardt we introduced a new spectral problem, the so-called generalized indefinite string. This spectral problem is quadratic in spectral parameter. As our first step, we developed the direct spectral theory for this new class of spectral problems. Our next main result is a solution of the inverse spectral problem for generalized indefinite strings. This solves a long-standing open problem going back to Krein and Langer and also extends the famous M. G. Krein spectral theory of strings to an indefinite setting. Finally, we demonstrated that this new spectral problem plays a key role in the study of the so-called conservative solutions to the Camassa-Holm equation, namely, it serves as an isospectral problem for the conservative Camassa-Holm flow in certain important cases.