Higher Order Approximation in the Reductive Perturbation Method. I. The Weakly Dispersive System

The higher order approximations in the reductive perturbation method are studied for the weakly dispersive nonlinear system. It is shown that the secular terms appearing in the high order terms are eliminated by adding to the Korteweg-de Vries equation the derivatives of the higher-order conserved densities, the physical effects of which are given by the renormalization of the velocities of the KdV solitons.