On the Integration of the Periodic Camassa–Holm Equation with an Integral-Type Source

In the present paper, we study the integration of the periodic Camassa–Holm equation with an integral-type 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. We show that the periodic Camassa–Holm equation with an integral-type source is also an important theoretical model as it is a completely integrable system. We obtain a representation for the solution of periodic Camassa–Holm equation with an integral-type source in the framework of the inverse spectral problem for a weighted Shturm–Liouville operator.