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Z. Naturforsch. 66a, 468 – 480
(2011)
doi:10.5560/ZNA.2011-0002
Near Integrability in Low Dimensional Gross–Neveu Models
Faculty of Mathematics and Informatics, Sofia University, 5 J. Bourchier Blvd., 1164 Sofia, Bulgaria
Received November 25, 2010 / revised March 19, 2011
Reprint requests to: O. C.; E-mail:
christov@fmi.uni-sofia.bg
The low-dimensional Gross–Neveu models are studied. For the systems, related to the Lie algebras so(4), so(5), sp(4), sl(3), we prove that they have Birkhoff-Gustavson normal forms which are integrable and non-degenerate in Kolmogorov–Arnold–Moser (KAM) theory sense. Unfortunately, this is not the case for systems with three degrees of freedom, related to the Lie algebras so(6) ~ sl(4), so(7), sp(6); their Birkhoff–Gustavson normal forms are proven to be non-integrable in the Liouville sense. The last result can easily be extended to higher dimensions.
Key words: Normal Forms; Kolmogorov–Arnold–Moser Theory; Non-Integrability.