Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation
There are very few completely integrable lattice differential equations. The list includes the Ablowitz--Ladik lattice, the Toda lattice, the Blaszak--Marciniak lattice, and their modifications. The authors construct two hierarchies of new lattice differential equations, which are related to the same discrete isospectral problem \[ \psi_{n+1} = U_n(u,\lambda) \psi_n, \quad U_n(u,\lambda) = \left( \begin{matrix} \lambda & 1 + r_n \\ \lambda s_n & 1 + s_n + r_n s_n \end{matrix} \right). \] The hierarchies are built in positive and negative powers of \(\lambda\) in the power series expansions. Not only integrable lattice equations are computed in the explicit form but also the Hamiltonian structure, the conservation laws, and the Darboux transformations are computed for equations of positive hierarchy.
scientific article; zbMATH DE number 5627006
scientific article