The isospectral problem for the Camassa-Holm shallow
water equation is a weighted Sturm-Liouville problem. The spectral
data consists of a continuous spectrum and a finite or infinite number
of eigenvalues. We present an approach for determining the evolution
of the scattering data, applicable independently of the number of
eigenvalues. In the case of finitely many eigenvalues the initial
value problem can be solved by the inverse
scattering method.