On the Poisson structure and action-angle variables for the complex modified Korteweg-de Vries equation

Abstract

In this paper, we employ the inverse scattering approach to study the Poisson structure and action-angle variables of the complex modified Korteweg-de Vries (cmKdV) equation. We first derive the cmKdV equation via the principle of variation. Then, we successfully obtain the Poisson brackets for the scattering data of the equation. Furthermore, 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.