Asymptotic irrelevance of the KdV hierarchy

Abstract:
All the equations in the KdV hierarchy share many features, as singular algebraic solutions, solitons, etc.  The reason for that is that these features are really already in the Schrödinger equation, the different KdV equations just prescribe diffrent speeds to reach the asymptotic regime. We can even calculat the phase shifts from inverse scattering without recourse to any KdV-type equations.