The KdV hierarchy and the propagation of solitons on very long distances

The Korteweg-de Vries (KdV) equation is first derived from a general system of partial differential equations. An analysis of the linearized KdV equation satisfied by the higher order amplitudes shows that the secular-producing terms in this equation are the derivatives of the conserved densities of KdV. Using the multi-time formalism, we prove that the propagation on very long distances is governed by all equations of the KdV hierarchy. We compute the soliton solution of the complete hierarchy, which allows to give a criterion for the existence of the soliton.