A Few Expanding Integrable Models, Hamiltonian Structures and Constrained Flows

Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer—Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especially, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and t n -constrained flows whose adjoint representations and the Lax pairs are given.