Why do soliton equations come in hierarchies?

Abstract

In this article, an identity satisfied by the so-called recursion operator is derived. The identity generates by itself an infinite sequence of Lax pairs, thus ensuring the complete integrability of the corresponding hierarchy of nonlinear evolution equations. It is also shown that this identity yields the familiar property that the squares of eigenfunctions of the associated linear spectral problem satisfy the linearized version of the respective soliton equation.