On the Inverse Spectral Problem for the Camassa–Holm Equation

Abstract

A key basis for seeking periodic solutions of the Camassa–Holm equation is to understand the associated spectral problemy′=14y+λmy.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 widthsλ2n−λ2n−1for a sequence 0<λ1⩽λ2<… of single or double eigenvalues tending to infinity is the real analyticity ofm. The case of a purely simple spectrum is typical of 0>m∈C1(R).