Communications in Mathematical Physics The Camassa-Holm Hierarchy , N-Dimensional Integrable Systems , and Algebro-Geometric Solution on a Symplectic Submanifold
Communications in Mathematical Physics The Camassa-Holm Hierarchy , N-Dimensional Integrable Systems , and Algebro-Geometric Solution on a Symplectic Submanifold
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发表时间：
2003
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通讯作者：
Z. Qiao
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作者：
Z. Qiao
本文证明了Camassa-Holm（CH）谱问题产生两个不同的非线性发展方程族，一个是负阶CH族，另一个是正阶CH族.通过求解一个关键的矩阵方程，这两个CH族都具有零曲率表示。我们看到著名的CH方程包含在负阶CH族中，而Dym型方程包含在正阶CH族中。进一步地，在势函数和本征函数之间的两个约束条件下，CH谱问题转化为：1。一个新的Neumann-like N维系统，当它被限制到R2 N的辛子流形中时，通过使用Dirac-Poisson括号和r-矩阵过程证明了它是可积的;和2.在全R2 N中考虑一个新的Bargmann型N维系统，利用标准Poisson括号和r-矩阵过程证明了该系统是可积的.本文提出了两种4 × 4代替N ×N的r-矩阵结构。一类是与负序CH谱系相关的Neumann类系统（非尖峰CH系统），另一类是与正序CH谱系相关的Bargmann类系统（也非尖峰CH系统）.证明了整个CH方程族（正、负阶的积分微分方程族）都有服从相应约束关系的参数解。特别地，约束在R2 N中辛子流形上的CH方程和Dym型方程都有参数解。此外，我们还发现CH方程的参数解与峰子的规范解是不等价的。通过求解CH方程在辛子流形上的参数表示，得到了CH方程的一类新的代数几何解。
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 R 2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process. In the paper, we present two 4 × 4 instead of N ×N r-matrix structures. One is for the Neumann-like system (not the peaked CH system) related to the negative order CH hierarchy, while the other one is for the Bargmann-like system (not the peaked CH system, either) related to the positive order hierarchy. The whole CH hierarchy (an integro-differential hierarchy, both positive and negative order) is shown to have the parametric solutions which obey the corresponding constraint relation. In particular, the CH equation, constrained to a symplectic submanifold in R2N , and the Dym type equation have the parametric solutions. Moreover, we see that the kind of parametric solution of the CH equation is not gauge equivalent to the peakons. Solving the parametric representation of the solution on the symplectic submanifold gives a class of a new algebro-geometric solution of the CH equation.