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.