Abstract
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 × 2 matrix spectral problems of the mKdV equation and obtain a pair of integrable Hamiltonian systems in 4 N dimensions. To get the integrabilities of the resulting 4 N-dimensional systems, we introduce a group of new 3 × 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 6 N dimensions. We show that the 4 N-dimensional integrable systems are just the restrictions of the 6 N-dimensional systems to a special symplectic submanifold.
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