Asymptotic Irrelevance of the KdV Hierarchy

All the equations of the KdV hierarchy share many common features, like single and multiple soliton solutions, singular algebraic potentials, etc.; in particular the phase shifts for solitonsoliton scattering are all the same in the chain KdV1, KdV2, etc., depending only on the momenta. We exhibit the reason for this behaviour: as these equations are originated as isopectral deformations of the Schrödinger equation in one dimension, the above features are really properties of the Schrödinger operator, and the different KdV equations just prescribe different “speeds”, given by the dispersion law, for reaching the asymptotic regime. In particular, we show how the “interaction” is sum of two-body forces, what is the key for the solubility of these equations in the first place, and then calculate the phase shifts from inverse scattering (as a double-Darboux factorization), obtaining the known result, independent of the hierarchy, without ever using any of the KdV equations. We also obtain the Hirota formula for N solitons as a Wronskian, by creating them one by one from the u(x) = 0 potential.