## A Lie algebra containing four parameters for the generalized Dirac hierarchy.(English)Zbl 1218.37087

Summary: A Lie algebra containing four parameters is obtained, whose commutation operation is concise, and the corresponding computing formula of constant $$\gamma$$ in the variational identity is presented in this paper. As application, a new Liouville integrable hierarchy which can be reduced to the Dirac hierarchy is derived by designing a special isospectral problem. We call it generalized Dirac hierarchy.

### MSC:

 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures 17B80 Applications of Lie algebras and superalgebras to integrable systems
Full Text:

### References:

 [1] Tu, G., A hierarchy of new integrable systems and its Hamiltonian structure, Sci. sinica (series A), 12, 1243, (1988) [2] Ma, W.X., Binary nonlinearization for the Dirac systems, Chin. ann. math. ser. B, 18, 79-88, (1997) · Zbl 0874.35105 [3] Ma, W.X., Multi-component bi-Hamiltonian Dirac integrable equations, Chaos, solitons & fractals, 39, 282-287, (2009) · Zbl 1197.37079 [4] Guo, F.; Zhang, Y., The quadratic-form identity for constructing the Hamiltonian structure of integrable systems, J. phys. A, 38, 8537, (2005) · Zbl 1077.37045 [5] Zhang, Y., A general boite – pempinelli – tu hierarchy and its bi-Hamiltonian structure, Phys. lett. A, 317, 3, 280, (2003) · Zbl 1027.37042 [6] Hu, X., A powerful approach to generate new integrable systems, J. phys. A, 27, 2497, (1994) · Zbl 0838.58018 [7] Zhang, Y., A subalgebra of loop algebra and its applications, Chin. phys., 13, 2, 132, (2004) [8] Fan, E., A Liouville integrable Hamiltonian system associated with a generalized kaup – newell spectral problem, Physica A, 301, 105, (2001) · Zbl 0977.37039 [9] Ma, W., Integrable couplings of soliton equations by perturbations I, a general theory and application to the KdV hierarchy, Meth. appl. anal., 7, 21, (2000) · Zbl 1001.37061 [10] Ma, W.; Xu, X.; Zhang, Y., Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. lett. A, 351, 125, (2006) · Zbl 1234.37049 [11] Ma, W.; Xu, X.; Zhang, Y., Semidirect sums of Lie algebras and discrete integrable couplings, J. math. phys., 47, 053501, (2006) · Zbl 1111.37059 [12] Ma, W.; Chen, M., Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras, J. phys. A: math. gen., 39, 10787, (2006) · Zbl 1104.70011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.