Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation

Abstract

A modified Boussinesq hierarchy associated with a 3×3 matrix spectral problem and its generalized Hamiltonian form are derived. A class of new finite-dimensional Hamiltonian systems are obtained with the help of the nonlinearization approach of Lax pairs. The generating function of the integrals is presented, by which the class of new finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. As an application, solutions of the modified Boussinesq equation are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.