KdV hierarchy

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

Let ${\displaystyle T}$ be translation operator defined on real valued functions as ${\displaystyle T(g)(x)=g(x+1)}$. Let ${\displaystyle {\mathcal {C}}}$ be set of all analytic functions that satisfy ${\displaystyle T(g)(x)=g(x)}$, i.e. periodic functions of period 1. For each ${\displaystyle g\in {\mathcal {C}}}$, define an operator ${\displaystyle L_{g}(\psi )(x)=\psi ''(x)+g(x)\psi (x)}$ on the space of smooth functions on ${\displaystyle \mathbb {R} }$. We define the Bloch spectrum ${\displaystyle {\mathcal {B}}_{g}}$ to be the set of ${\displaystyle (\lambda ,\alpha )\in \mathbb {C} \times \mathbb {C} ^{*}}$ such that there is a nonzero function ${\displaystyle \psi }$ with ${\displaystyle L_{g}(\psi )=\lambda \psi }$ and ${\displaystyle T(\psi )=\alpha \psi }$. The KdV hierarchy is a sequence of nonlinear differential operators ${\displaystyle D_{i}:{\mathcal {C}}\to {\mathcal {C}}}$ such that for any ${\displaystyle i}$ we have an analytic function ${\displaystyle g(x,t)}$ and we define ${\displaystyle g_{t}(x)}$ to be ${\displaystyle g(x,t)}$ and ${\displaystyle D_{i}(g_{t})={\frac {d}{dt}}g_{t}}$, then ${\displaystyle {\mathcal {B}}_{g}}$ is independent of ${\displaystyle t}$.

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