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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.
A. Constantin, V. Gerdjikov and R. Ivanov, Inverse Scattering Transform for the Camassa-Holm equation, Inv. Problems 22 (2006) 2197-2207