The semi-discrete AKNS system: Conservation laws, reductions and continuum limits

In this paper, the semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy is shown in spirit composed by the Ablowitz-Ladik flows under certain combinations. Furthermore, we derive its explicit Lax pairs and infinitely many conservation laws, which are non-trivial in light of continuum limit. Reductions of the semi-discrete AKNS hierarchy are investigated to include the semi-discrete Korteweg-de Vries (KdV), the semi-discrete modified KdV, and the semi-discrete nonlinear Schr\"odinger hierarchies as its special cases. Finally, under the uniform continuum limit we introduce in the paper, the above results of the semi-discrete AKNS hierarchy, including Lax pairs, infinitely many conservation laws and reductions, recover their counterparts of the continuous AKNS hierarchy.