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.^[1]^[2]

References

↑ Fabio A. C. C. Chalub1 and Jorge P. Zubelli, "Huygens’ Principle for Hyperbolic Operators and Integrable Hierarchies"
↑ 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

Sources

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

KdV hierarchy at the Dispersive PDE Wiki.

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[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

KdV hierarchy

Details

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References

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External links