On the integration of the periodic Camassa–Holm equation with a self-consistent source

Recently, much attention has been paid to non-linear equations with a self-consistent source that have soliton solutions. Sources arise in solitary waves with a variable speed and lead to a variety of physical models. Such models are usually used to describe interactions between solitary waves. The Cauchy problem for the Camassa–Holm equation with a source in the class of periodic functions is considered in this paper. The main result of this work is a theorem on the evolution of the spectral data of the weighted Sturm–Liouville operator where potential of the operator is a solution of the periodic Camassa–Holm equation with a source. The obtained relations allow one to apply the method of the inverse spectral transform to solve the Cauchy problem for the periodic Camassa–Holm equation with a source.