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. In 1965, Zabusky and Kruskal discovered that the pulselike solitary wave solution of the KdV equation had a property which was previously unknown: two solitary waves preserve their shape and size after interaction. They hence termed these solutions solitons. Shortly after this discovery, 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 indeed worked for another physically significant nonlinear evolution equation, namely, the nonlinear Schrödinger equation. However, for sufficiently regular initial conditions, the solutions of the KdV equation are global, whereas it is known that some shallow water waves break. 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 weave breaking. In particular, as first noted by Camassa and Holm, wave breaking may occur if the associated Sturm-Liouville spectral 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, i.e., the density changes sign. The aims of the project are to develop direct and inverse scattering theory for the Sturm-Liouville equation with an indefinite density function and the study of the blow up phenomena for the CH equation with the help of the inverse scattering transform.