The quadratic-form identity for constructing the Hamiltonian structure of integrable systems. (English) Zbl 1077.37045

Guo, Fukui; Zhang, Yufeng

J. Phys. A, Math. Gen. 38, No. 40, 8537-8548 (2005).

Summary: A usual loop algebra, not necessarily the matrix form of the loop algebra \(\widetilde A_{n-1}\), is also made use of for constructing linear isospectral problems, whose compatibility conditions exhibit a zero-curvature equation from which integrable systems are derived. In order to look for the Hamiltonian structure of such integrable systems, a quadratic-form identity is created in the present paper whose special case is just the trace identity; that is, when taking the loop algebra \(\widetilde A_1\), the quadratic-form identity presented in this paper is completely consistent with the trace identity.

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B80	Applications of Lie algebras and superalgebras to integrable systems

Keywords:

m-AKNS hierarchy; loop algebra; linear isospectral problems; trace identity