Communications in Mathematical Physics

, Volume 239, Issue 1–2, pp 309–341

The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold

Article

DOI: 10.1007/s00220-003-0880-y

Cite this article as:
Qiao, Z. Commun. Math. Phys. (2003) 239: 309. doi:10.1007/s00220-003-0880-y

Abstract

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ℝ2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ℝ2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Los Alamos National LaboratoryLos AlamosUSA
  2. 2.Institute of MathematicsFudan UniversityShanghaiP.R. China

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