Integration of Camassa-Holm equation with a self-consistent source of integral type

Abstract

The work is devoted to studying Camassa-Holm equation with a self-consistent source of integral type. The source of the consistent equation corresponds to the continuous spectrum of a spectral problem related with the Camassa-Holm equation. As it is known, integrable systems admit operator Lax representation $L_t = [L,A]$, where $L$ is a linear operator, while $A$ is some skew-symmetric operator acting in a Hilbert space. A generalized Lax representation for the considered equation is of the form $L_t = [L,A]+C$, where $C$ is the sum of differential operators with coefficients depending on solutions of spectral problems for the operator $L$. The construction of self-consistent source for the considered operator is based on the fact that exactly squares of eigenfunctions of the spectral problems are essential while solving integrable equations by the inverse scattering transform. Moreover, for the considered type of equations the evolution of the eigenfunctions in the generalized Lax representation has a singularity. The application of the inverse scattering transform is based on the spectral problem related with the classical Camassa-Holm equation. We describe the evolution of scattering data of this spectral problem with a potential being a solution of the Camassa-Holm equation with a self-consistent source. While describing the evolution of the spectral data, we employ essentially Sokhotski-Plemelj formula. The results of the work on the evolution of the scattering data related with the discrete spectrum are based on the methods used in the previous works by the authors. The obtained results, formulated as a main theorem, allow us to apply the inverse scattering transform for solving the Cauchy problem for the considered equation. Our technique can be easily extended to higher analogues of the Camassa-Holm equation.