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[]
- ^ 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.
- ^ 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, vol. 79, Cambridge University Press, ISBN 978-0-521-75307-4, MR 1992536
External links[]
- KdV hierarchy at the Dispersive PDE Wiki.
Categories:
- Partial differential equations
- Solitons
- Exactly solvable models