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 ${\displaystyle T(g)(x)=g(x+1)}$. Let ${\mathcal {C}}$ 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 $\mathbb {R}$. 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 $\psi$ 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 $i$ we have an analytic function $g(x,t)$ and we define ${\displaystyle g_{t}(x)}$ to be $g(x,t)$ and ${\displaystyle D_{i}(g_{t})={\frac {d}{dt}}g_{t}}$, then ${\displaystyle {\mathcal {B}}_{g}}$ is independent of $t$.

The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.^[1][2]

See also

• Witten's conjecture
• Huygens' principle

References

1. Fabio A. C. C. Chalub1 and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"
2. Yuri Yu. Berest and Igor M. Loutsenko, "Huygens’ Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation", arXiv:solv-int/9704012 DOI 10.1007/s002200050235

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

External links

• at the Dispersive PDE Wiki.