  An independent derivation of solutions to the Camassa-Holm equation in terms
of multi-dimensional theta functions is presented using an approach based on
Fay's identities. Reality and smoothness conditions are studied for these
solutions from the point of view of the topology of the underlying real
hyperelliptic surface. The solutions are studied numerically for concrete
examples, also in the limit where the surface degenerates to the Riemann
sphere, and where solitons and cuspons appear.
