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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On the Complete Integrability of some
Lax Equations on a Periodic Lattice


Author: Luen-Chau Li
Journal: Trans. Amer. Math. Soc. 349 (1997), 331-372
MSC (1991): Primary 58F07; Secondary 58F05, 65F15
DOI: https://doi.org/10.1090/S0002-9947-97-01729-7
MathSciNet review: 1373643
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Abstract: We consider some Lax equations on a periodic lattice with $N=2$ sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.


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Additional Information

Luen-Chau Li
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: luenli@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01729-7
Received by editor(s): May 16, 1994
Received by editor(s) in revised form: October 6, 1995
Article copyright: © Copyright 1997 American Mathematical Society