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**An extended trace identity and applications.**
*(English)*
Zbl 1172.70009

From the abstract: For the loop algebras in the form of non-square matrices, their commuting operations can be used to set up linear isospectral problems. In order to look for Hamiltonian structures of the corresponding integrable evolution hierarchies of equations, an extended trace identity is obtained by means of commutators, which undoes the constraint on the known trace identity proposed by G. Tu [J. Math. Phys. 30, No. 2, 330–338 (1989; Zbl 0678.70015)], and has an obvious simplicity in applications compared with the quadratic-form identity given by F. Guo and Y. Zhang [J. Phys. A, Math. Gen. 38, No. 40, 8537–8548 (2005; Zbl 1077.37045)].

Reviewer: Marian Ioan Munteanu (Iaşi)

### MSC:

70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |

37N05 | Dynamical systems in classical and celestial mechanics |

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\textit{F. Guo} and \textit{Y. Zhang}, Chaos Solitons Fractals 36, No. 4, 1113--1119 (2008; Zbl 1172.70009)

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### References:

[1] | Guizhang, Tu, A new integrable system and its Hamiltonian structure, Chin Sci (A Ser), 12, 1243-1252 (1988) |

[2] | Guizhang, Tu, The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J Math Phys, 30, 2, 330-338 (1989) · Zbl 0678.70015 |

[3] | Guizhang, Tu, On Liouville integrability of zero curvature equations and the Yang hierarchy, J Phys A, 22, 2375-2392 (1989) · Zbl 0697.58025 |

[4] | Fukui, Guo; Yufeng, Zhang, A types of AKNS integrable model, Acta Phys Sin, 51, 5, 951-954 (2002) · Zbl 1202.37084 |

[5] | Fukui, Guo; Yufeng, Zhang, A new loop algebra and a corresponding integrable hierarchy as well as its integrable couplings, J Math Phys, 44, 12, 5793-5803 (2003) · Zbl 1063.37068 |

[6] | Fukui, Guo; Yufeng, Zhang, A new loop algebra and its applications, Chaos, Solitons & Fractals, 22, 5, 1063-1069 (2004) · Zbl 1055.37068 |

[7] | Fukui, Guo; Yufeng, Zhang, The quadratic-form identity for constructing the Hamiltonian structure of integrable systems, J Phys A, 38, 8537-8548 (2005) · Zbl 1077.37045 |

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