New semi-discrete AKNS hierarchy and its reductions

Abstract

In the paper we consider the new semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchies and their reduction cases. In terms of a new linear combination of the two-potential Ablowitz–Ladik (AL) recursion operator and inverse recursion operator, the new sdAKNS isospectral and non-isospectral flows, as well as corresponding hierarchies are derived. By means of the continuous limit theory, both these flows and hierarchies go to the continuous AKNS flows and hierarchies, and the recursion operator goes to the AKNS recursion operator. Moreover, as the reduction systems of the new sdAKNS system, the vector form of the new semi-discrete nonlinear Schrödinger (sdNLS) hierarchies and the new semi-discrete modified Korteweg–de Vries (sdmKdV) hierarchies can be deduced under the different reductions. Furthermore, a new sdNLS equation and a new sdmKdV equation are derived. The above equations go to the counterparts of the continuous AKNS system and its reduction cases under continuous limit.