An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach 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. We therefore have to develop different asymptotic expansions.MSC: 35P25, 35Q15, 35Q35, 35Q51, 35Q53

The direct and the inverse scattering problem for the first order linear systems of the type Lψ(x,λ)≡(i(d/dx)+q(x)−λJ)ψ(x,λ)=0, J∈𝔥, q(x)∈𝔤J , which generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals, and Coifman (CBC) is analyzed herein. Here J is a regular complex constant element of the Cartan subalgebra 𝔥⊆𝔤 of the simple Lie algebra 𝔤 and the potential q(x) vanishes fast enough for ‖x‖ → ∞ taking values in the image 𝔤J of adJ. The CBC results are generalized and the fundamental analytic solution m(x,λ) for any choice of the irreducible finite-dimensional representation V of 𝔤 is constructed. Four pairwise equivalent minimal sets of scattering data for L, invariant with respect to the choice of the representation of 𝔤, are extracted from the asymptotics of m(x,λ) for x → ±∞. From m(x,λ) the resolvent of L is constructed in the adjoint representation Vad and the completeness relation is proven for the eigenfunctions of L in Vad. It is also proven that the discrete spectrum of L consists of the sets of zeroes of certain spectral invariants D+j(λ) of L.

The direct and the inverse scattering problems for the generalised Zakharov-Shabat spectral problem L related to a given semi-simple Lie algebra are solved. The fundamental analytic solutions and the minimal set of scattering data F are constructed. The mapping from the set of potentials to F is shown to be a generalised Fourier transform over the sets ( Psi ), ( Phi ) of the adjoint solutions of L. The expansions for the potential of L and its variations over ( Psi ), ( Phi ) are obtained and used to investigate the corresponding non-linear evolution equations (NLEE) on homogeneous spaces. These are shown to be spectral expansions for the generating operators Lambda +or- which play a fundamental role in the theory of the NLEE. The approach is explicitly gauge covariant, which allows one to investigate also the corresponding gauge-equivalent NLEE.

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