KdV hierarchy

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

Let be translation operator defined on real valued functions as . Let be set of all analytic functions that satisfy , i.e. periodic functions of period 1. For each , define an operator on the space of smooth functions on . We define the Bloch spectrum to be the set of such that there is a nonzero function with and . The KdV hierarchy is a sequence of nonlinear differential operators such that for any we have an analytic function and we define to be and , then is independent of .

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

See also

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

External links

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