Summary: A key basis for seeking periodic solutions of the Camassa-Holm equation
is to understand the associated spectral problem .
The periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum. The potential can then be reconstructed from the periodic spectrum. A necessary and sufficient condition for exponential decrease of the widths for a sequence of single or double eigenvalues tending to infinity is the real analyticity of . The case of a purely simple spectrum is typical of .