The Korteweg-de Vries hierarchy: structure and solution

We attempt to realize the structure of the Korteweg-de Vries (KdV) hierarchy by using a simple dimensional analysis. The specific results presented refer to equations of the hierarchy and conserved Hamiltonian densities associated with them. Based on the Gel’fand-Levitan-Marchenko equation we construct a series expansion for the unstable solution of the KdV-like equations by using the continuous part of the spectrum for the Schrödinger operator. Our result is in exact agreement with that obtained from the Sabatier’s formulation of the inverse problem for rational reflection coefficients.