Abstract

The inverse scattering method is developed for the Camassa–Holm equation. As an illustration the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. This requires different asymptotic expansions for the eigenfunctions.

The squared eigenfunctions of the spectral problem associated with the Camassa–Holm equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the Camassa–Holm hierarchy as a generalized Fourier transform. There exists a completeness relation for the squared solutions of the Camassa–Holm spectral problem. We show that all the fundamental properties of the Camassa–Holm equation such as the integrals of motion, the description of the equations of the whole hierarchy and their Hamiltonian structures can be naturally expressed making use of the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions.