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 $(\lambda,\alpha) \in \mathbb{C}\times\mathbb{C}^*$ 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

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.

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KdV hierarchy

See also

References

External links