On the Camassa-Holm hierarchy and N-dimensional integrable systems

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. Associated with the two hierarchies, a new Neumann-like and a new Bargmann-like N -dimensional integrable system are presented. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We find that the well-known CH equation is included in the negative order CH hierarchy while a Dym type equation is included in the positive order CH hierarchy. Furthermore, under some constraint conditions between eigenfunctions and potential the CH spectral problem yields: 1. a new Neumann-like 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