On the continuous limit of integrable lattices I. The KM lattice and KdV theory

From the MR review by Malcolm Adams " It has been known for some time that the Korteweg-de Vries (KdV) equation can be obtained as a continuous limit of the Kac-Moerbeke (KM) lattice system, an integrable system on a one-dimensional lattice of particles related to the Toda lattice [M. Schwarz, Jr., Adv. in Math. 44 (1982), no. 2, 132--154; MR0658538 (83i:35152)]. Although that work helps to give insight into the structure of the KdV equation, it does not show that the entire KdV hierarchy of commuting evolutionary partial differential equations can be obtained in the same limiting fashion." The present paper fills that gap by giving a systematic construction of the entire KdV theory as a continuous limit of the KM theory. The approach is first to give the correspondence between the bi-Hamiltonian structures of the two theories and then to produce the corresponding hierarchies through recursion operators. The paper is well written and fairly self-contained, giving a nice brief overview of KM theory.