KdV hierarchy

In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which starts with the Korteweg–de Vries equation.

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 $t$.

External link

• KdV hierarchy at the Dispersive PDE Wiki.