A generalized MKdV hierarchy, tri-Hamiltonian structure, higher-order binary constrained flows and its integrable couplings system

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A subalgebra of higher-dimension loop algebra A 2 ˜ is constructed, from which a new 3 × 3 isospectral problem is designed. By making use of Tu’s scheme, an integrable Hamiltonian hierarchy of equations in the sense of Liouville is obtained, which possesses tri-Hamiltonian structure. As reduction cases of the hierarchy presented in this paper, the generalized MKdV equation is engendered. By establishing binary symmetric constraints, three constrained flows of the hierarchy are presented, which are then reduced to Hamiltonian systems. Finally, an integrable coupling system is obtained by constructing a high-dimension loop algebra.