An alternative derivation of the pulse solutions of the KdV hierarchy

Abstract

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.