Hamiltonian structure of discrete soliton systems

We describe an approach for investigating the Hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem. By using the so-called implicit representations of the isospectral flows, we demonstrate the existence of the recursion operator L , which is a strong and hereditary symmetry of the flows. It is then proven that every equation in the isospectral hierarchy possesses the Hamiltonian structure if L has a skew-symmetric factorization and the first equation ( u t = K (0) ) in the hierarchy satisfies some simple condition. We obtain related properties, such as the implectic-symplectic factorization of L , the Liouville complete integrability and the multi-Hamiltonian structures of the isospectral hierarchy. Four examples are given.