Modified r-matrix and separation of variables for the modified Korteweg-de Vries (MKdV) hierarchy

In this article, it is first given that a Lax representation in terms of 2 × 2 matrices for the completeey integrable finite-dimensional Hamiltonian system (CIFHS) (H) produced through the nonlinearization procedure for the MKdV hierarchy, and then an associated non-dynamical modified r-matrix is constructed. By making use of this r-matrix and matrix trace equality, a set of finite-dimensional involutive functions Fm (m = 0, 1,…,F0 = H), which guarantees the integrability of Hamiltonian systems (H), and the Lax representations in terms of 2 × 2 matrices for the whole Hamiltonian hierarchies (Fm) (m = 0, 1,…) are obtained. Moreover, the involutive solutions of the MKdV hierarchy are given. Finally, it is found that the Hamiltonian-Jacobi equation for the Hamiltonian system (H) can be separable under a group of new coordinates introduced by the 2 × 2 lax matrix.