KdV hierarchy

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

Details

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 $(\lambda ,\alpha )\in \mathbb {C} \times \mathbb {C} ^{*}$ such 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 $D_{i}:{\mathcal {C}}\to {\mathcal {C}}$ such 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 $t$.

The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.

See also

• Witten's conjecture
• Huygens' principle

Sources

• Gesztesy, Fritz; Holden, Helge (2003), Soliton equations and their algebro-geometric solutions. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 79, Cambridge University Press, ISBN 978-0-521-75307-4, MR 1992536

External links

• KdV hierarchy at the Dispersive PDE Wiki.