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

Li, Luen-Chau

doi:10.1090/S0002-9947-97-01729-7

On the Complete Integrability of some Lax Equations on a Periodic Lattice
HTML articles powered by AMS MathViewer

by Luen-Chau Li PDF

Trans. Amer. Math. Soc. 349 (1997), 331-372 Request permission

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.

References

M. J. Ablowitz and J. F. Ladik, Nonlinear differential-difference equations and Fourier analysis, J. Mathematical Phys. 17 (1976), no. 6, 1011–1018. MR 427867, DOI 10.1063/1.523009
Mark Adler and Pierre van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math. 38 (1980), no. 3, 318–379. MR 597730, DOI 10.1016/0001-8708(80)90008-0
Moody Chu, A continuous approximation to the generalized Schur decomposition, Linear Algebra Appl. 78 (1986), 119–132. MR 840171, DOI 10.1016/0024-3795(86)90019-4
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
B. A. Dubrovin, Igor Moiseevich Krichever, and S. P. Novikov, Integrable systems. I, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 179–284, 291 (Russian). MR 842910
P. A. Deift and L. C. Li, Generalized affine Lie algebras and the solution of a class of flows associated with the $QR$ eigenvalue algorithm, Comm. Pure Appl. Math. 42 (1989), no. 7, 963–991. MR 1008798, DOI 10.1002/cpa.3160420704
P. Deift, L. C. Li, and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 42 (1989), no. 4, 443–521. MR 990138, DOI 10.1002/cpa.3160420405
P. Deift, L. C. Li, T. Nanda, and C. Tomei, The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39 (1986), no. 2, 183–232. MR 820068, DOI 10.1002/cpa.3160390203
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
Otto Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York-Berlin, 1981. Translated from the German by Bruce Gilligan. MR 648106, DOI 10.1007/978-1-4612-5961-9
L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 905674, DOI 10.1007/978-3-540-69969-9
Phillip A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations, Amer. J. Math. 107 (1985), no. 6, 1445–1484 (1986). MR 815768, DOI 10.2307/2374412
I. M. Gel′fand and L. A. Dikiĭ, Fractional powers of operators, and Hamiltonian systems, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 13–29 (Russian). MR 0433508
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
Luen Chau Li, On the complete integrability of some Lax systems on $\textrm {GL}(n,\textbf {R})\times \textrm {GL}(n,\textbf {R})$, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 487–493. MR 1028140, DOI 10.1090/S0273-0979-1990-15961-0
Luen Chau Li, Two results on a class of Poisson structures on Lie groups, Math. Z. 210 (1992), no. 2, 327–337. MR 1166529, DOI 10.1007/BF02571801
—, The SVD flows on generic symplectic leaves are completely integrable, to appear in Adv. Math..
Luen Chau Li and Serge Parmentier, A new class of quadratic Poisson structures and the Yang-Baxter equation, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 6, 279–281 (English, with French summary). MR 956823
David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651, DOI 10.1007/978-1-4899-2843-6
C. B. Moler and G. W. Stewart, An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal. 10 (1973), 241–256. MR 345399, DOI 10.1137/0710024
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
A. G. Reyman and M. A. Semenov-Tian-Shansky, Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. II, Invent. Math. 63 (1981), no. 3, 423–432. MR 620678, DOI 10.1007/BF01389063
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
Peter Grassberger and Itamar Procaccia, Measuring the strangeness of strange attractors, Phys. D 9 (1983), no. 1-2, 189–208. MR 732572, DOI 10.1016/0167-2789(83)90298-1
Michael A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260. MR 842417, DOI 10.2977/prims/1195178514
Victor Vinnikov, Complete description of determinantal representations of smooth irreducible curves, Linear Algebra Appl. 125 (1989), 103–140. MR 1024486, DOI 10.1016/0024-3795(89)90035-9
B. L. van der Waerden, Algebra. Vol 1, Frederick Ungar Publishing Co., New York, 1970. Translated by Fred Blum and John R. Schulenberger. MR 0263582
Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. MR 723816