Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .