Siberian Federal University

Recently, much attention has been paid to the soliton equations with a self-consistent source. Physically, sources arise in solitary waves with a variable speed and lead to a variety of dynamics of physical models. With regard to their applications, these kinds of systems are usually used to describe interactions between different solitary waves. In this paper, we consider the Cauchy problem for the Camassa—Holm equation with a source in the class of periodic functions. The main result of this work is a theorem on the evolution of the spectral data of the weighted Sturm—Liouville operator whose potential is a solution to the periodic Camassa–Holm equation with a source. The obtained equalities allow us to apply the method of the inverse spectral transform to solve the Cauchy problem for the periodic Camassa-Holm equation with a source