Bi-Integrable Couplings of a Nonsemisimple Lie Algebra by Toda Lattice Hierarchy

W. X. Ma established theory of bi-integrable couplings to construct Hamilton structure of continuous bi-integrable couplings. In our paper, the theory of bi-integrable couplings is generalized to the discrete case. First, based on semi-direct sums of Lie subalgebra, a class of higher-dimensional 6×6 matrix Lie algebras is constructed. Moreover, starting from a new 6-order matrix spectral problem with a parameter, the bi-integrable couplings of the Toda lattice hierarchy was obtained from the proposed nonsemisimple higher-dimensional Lie algebras. Finally, the obtained discrete bi-integrable coupling systems are all written into their bi-Hamiltonian forms by the discrete variational identity.