The Zabusky-Kruskal lattice (ZK) was derived as a finite-difference approximation of the Korteweg-de Vries equation (KdV) for the purpose of numerical simulation of that PDE. Like the Fermi-Pasta-Ulam (FPU) lattice from which it was ultimately derived, ZK was also observed to exhibit near-recurrence of its initial state at regular time intervals.
The recurrence in ZK, and more generally in nonlinear lattices, has not been completely explained, though it has been attributed to the solitons or, less specifically, the integrability of the KdV continuum limit. Such attribution leads naturally to the hypotheses that simulations on smaller grid separations and discretizations that are integrable as spatially discrete systems should exhibit stronger recurrences than observed in the original simulations of ZK. However, systematic simulations over a range of grid scales are not consistent with these hypotheses.
For the ZK and a finite-diffrence integrable discretization of KdV, recurrence of a low-mode initial state is observed to be strongest and most persistent at a middle scale. As grid scales shorten, repeated simulations show convergence (presumably to KdV), but recurrence intensity and persistence weaken. At the other end, for grid scales over a threshold value, there is no recurrence. Instead, for grid sizes beyond the threshold, energy is rapidly distributed across the spatial Fourier spectrum.