The reductive perturbation method and the Korteweg-de Vries hierarchy

Abstract

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variablesτ1,τ3,τ5, ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitudeζ0 satisfies the KdV equation in the timeτ3, it must satisfy the (2n+1)th order equation of the KdV hierarchy in the timeτn+1, withn=2,3,4, ... As a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms (ζ1,ζ2, ...) of the amplitude can be eliminated whenζ0 is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitudeζ0 satisfies the (2n+ 1)th order equation of the KdV hierarchy in the timeτ2n+1. This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.