Abstract:
The Inverse Scattering Transform is a powerful tool for solving initial value problems for certain nonlinear partial differential equations. Originally developed for the Korteweg--de Vries equation, this method has since been successfully extended to various other completely integrable systems. I will review this way of solution and show how it can be employed to integrate the Camassa--Holm equation, which arises as a model for shallow water waves. This approach depends on the solution of an inverse spectral problem for inhomogeneous vibrating strings with indefinite mass distributions.