KdV hierarchy

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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.

See also

  • Witten's conjecture

References

  • 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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