The Reductive Perturbation Method and the Korteweg—de Vries Hierarchy

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variables τ 1, τ 3, τ 5, …, 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 amplitude ζ 0 satisfies the KdV equation in the time τ 3, it must satisfy the (2n +1)th order equation of the KdV hierarchy in the time τ 2n+ 1, with n = 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 (ζ 1,ζ 2,…) of the amplitude can be eliminated when ζ 0 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 amplitude ζ 0 satisfies the (2n+1)th order equation of the KdV hierarchy in the time τ 2n+ 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.