The multi-component dispersive long wave equation hierarchy, its integrable couplings and their Hamiltonian structures

An isospectral problem is established based on loop algebra X∼, by making use of the generalized Tu scheme the multi-component Liouville integrable dispersive long wave (DLW) equation hierarchy is obtained. Then, two expanding loop algebra F∼MandY∼ are presented, which devoted to working out two integrable couplings of the multi-component DLW equation hierarchy. Finally, the Hamiltonian structures of the multi-component DLW equation hierarchy and the second integrable couplings are obtained by employing the trace variational identity.