Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators
Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators
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DOI：
10.1016/j.amc.2011.05.103
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发表时间：
2011-06
期刊：
影响因子：
--
通讯作者：
Yuqin Yao;Yehui Huang;Yuan Wei;Yunbo Zeng
中科院分区：
文献类型：
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作者：
Yuqin Yao;Yehui Huang;Yuan Wei;Yunbo Zeng
当双哈密顿系统的两个哈密顿算子都是纯微分算子时，我们表明从[10]中的库珀什密特变形发展而来的广义库珀什密特变形（GKD）提供了一种从双哈密顿系统开始构造新的可积系统的有用方法。我们在 Harry Dym 层次、经典 Boussinesq 层次和耦合 KdV 层次的情况下，通过广义 Kupershmidt 变形构造了一些新的可积系统。我们证明了Harry Dym方程的GKD、经典Boussinesq方程的GKD和耦合KdV方程的GKD等价于这些具有自洽源的孤子方程的新的可积Rosochatius变形。我们为这些新系统提供了 Lax 对。因此，广义库珀什密特变形提供了一种从双哈密尔顿系统构造新可积系统的新方法，也提供了一种获得自洽源孤子方程Rosochatius变形的新方法。
When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.