Recurrence in the Korteweg-de Vries equation?

The Zabusky-Kruskal lattice (ZK) was derived as a finite-difference approximation of the Korteweg-de Vries equation for the purpose of numerical simulation of that PDE. Like the Fermi-Pasta-Ulam 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 has not been completely explained, though it has been attributed to the solitons or, less specifically, the integrability of the KdV continuum limit The attribution of recurrence to the integrablity of the continuum limit (i.e., KdV) leads naturally to the hypotheses that simulations (i) on smaller grid separations, (ii) with discretizations that are integrable as spatially discrete systems or (iii) with higher-order discretizations should all exhibit stronger recurrences than observed in the original simulations of ZK. However, systematic simulations over a range of grid scales with ZK, as well as an integrable discretization introduced here and a spectral discretization of KdV are not consistent with these hypotheses. On the contrary, for the ZK and an integrable finite-difference discretization of KdV, recurrence of a low-mode initial state is observed to be strongest and most persistent at an intermediate scale. We conclude that the observed recurrences is a lattice property and not a reflection of the integrable dynamics of KdV.