A Finite-Dimensional Completely Integrable System Associated with Boussinesq Hierarchy

In this paper, a new completely integrable system related to the complex spectral problem -φxx + (i/4)uφx + (i/4)(uφ)x + (1/4)vφ = iλφx and the constrained flows of the Boussinesq equations are generated. According to the viewpoint of Hamiltonian mechanics, the Euler–Lagrange equations and the Legendre transformations, a reasonable Jacobi–Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potential u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hierarchy are given.