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

[摘要] 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 Schrodinger 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 epsilon. We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of epsilon. Finally, by performing an appropriate resealing, we compare our results to those in the famous Zabusky and Kruskal's paper, showing very good agreement with the numerical results. (C) 2016 Elsevier B.V. All rights reserved.

[发布日期] 2016-10-15

[效力级别]  Proceedings Paper

[关键词] Korteweg-de Vries equation;Inverse scattering;WKB method;Solitons