A discrete multiple scales analysis of a discrete version of the korteweg-de Vries equation

A more elaborate discrete multiple scales analysis than that used by Newell in 1977 is performed on the Zabusky-Kruskal discretization of the Korteweg-de Vries (KdV) equation. This eventually leads to a set of partial difference equations describing the modulational behavior of a small harmonic wave modulated by a slowly varying envelope. In the case of certain modes of the carrier wave, the multiple scales analysis breaks down, indicating that in these cases the numerical solution deviates in behavior from that of the KdV equation. Numerical experiments are reported which confirm this.