On inverse scattering approach and action-angle variables to the Harry-Dym equation

Abstract

In this work, we employ the inverse scattering approach to study the Poisson structure and action-angle variables for the Harry-Dym equation. The Poisson brackets for the scattering data are presented. In consequence, the action-angle variables are expressed in terms of the scattering data. Interestingly, our results show that the coordinate expression and the spectral parameter expression of the Hamiltonian can be related by the conservation laws. Moreover, we establish the Wronskian relations of the Jost solution and the completeness relation for the squared solutions of the spectral problem, and prove that any function that decays rapidly at infinity can be expanded by the squared solutions.