KdV hierarchy

Let :$T$ be translation operator defined on real valued functions as :$T(g)(x)=g(x+1)$. Let :${\mathcal {C}}$ be set of all Analytic Functions that satisfy :$T(g)(x)=g(x)$, i.e. periodic functions of period 1. For each :$g\in {\mathcal {C}}$, define an operator

$L_{g}(\psi )(x)=\psi ''(x)+g(x)\psi (x)$

on the space of smooth functions on :$\mathbb {R}$. We define the Bloch spectrum :${\mathcal {B}}_{g}$ to be the set of :$(\lambda ,\alpha )\in \mathbb {C} \times \mathbb {C} ^{*}$ so that there is a nonzero function :$\psi$ with :$L_{g}(\psi )=\lambda \psi$ and :$T(\psi )=\alpha \psi$. The KdV hierarchy is a sequence of nonlinear differential operators  :$D_{i}:{\mathcal {C}}\to {\mathcal {C}}$ so that for any :$i$ we have an analytic function :$g(x,t)$ and we define :$g_{t}(x)$ to be :$g(x,t)$ and

$D_{i}(g_{t})={\frac {d}{dt}}g_{t}$,

then :${\mathcal {B}}_{g}$ is independent of :$t$.