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 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. Chalub, Fabio A. C. C.; Zubelli, Jorge P. (2006). "Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies". Physica D: Nonlinear Phenomena. 213 (2): 231–245. doi:10.1016/j.physd.2005.11.008.
  2. Berest, Yuri Yu.; Loutsenko, Igor M. (1997). "Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation". Communications in Mathematical Physics. 190 (1): 113–132. 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
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