Two kinds of new integrable decompositions of the mKdV equation

Two kinds of integrable decompositions of the mKdV equation are presented. We apply a new kind of binary nonlinearization approach of the spectral problem to the well-known 2×22×2 matrix spectral problems of the mKdV equation and obtain a pair of integrable Hamiltonian systems in 4N dimensions. To get the integrabilities of the resulting 4N  -dimensional systems, we introduce a group of new 3×33×3 matrix spectral problems for the mKdV hierarchy and apply the traditional binary nonlinearization approach of the spectral problem to them. As a result, we obtain another pair of integrable Hamiltonian systems in 6N dimensions. We show that the 4N-dimensional integrable systems are just the restrictions of the 6N-dimensional systems to a special symplectic submanifold.