Action-angle variables for the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation

Abstract

In this work, we present two finite-dimensional Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation by using the nonlinearization method. Moreover, the separation of variables on the common level set of Casimir functions is introduced to study these systems which are associated with a non-hyperelliptic algebraic curve. Finally, in light of the Hamilton–Jacobi theory, the action-angle variables for these systems are constructed, and the Jacobi inversion problem associated with the Hirota–Satsuma modified Boussinesq equation is obtained.