Small dispersion limit of the Korteweg-de Vries equation with periodic initial conditions and analytical description of the Zabusky-Kruskal experiment

We study the small dispersion limit of the Korteweg-de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky-Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the time-independent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ϵ. We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ϵ. Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal's paper, showing very good agreement with the numerical results.