Abstract
We study an iso-spectral deformation of the general matrix which is a natural generalization of the nonperiodic Toda lattice equation. This deformation is equivalent to the Cholesky flow, a continuous version of the Cholesky algorithm, introduced by Watkins. We prove the integrability of the deformation and give an explicit formula for the solution to the initial value problem. The formula is obtained by generalizing the orthogonalization procedure of Szegö. Using the formula, the solution to the LU matrix factorization can the constructed explicitly. Based on the root spaces for simple Lie algebras, we consider several reductions of the equation. This leads to generalized Toda equations related to other classical semi-simple Lie algebras which include the integrable systems studied by Bogoyavlensky and Kostant. We show these systems can be solved explicitly in a unified way. The behaviors of the solutions are also studied. Generically, there are two types of solutions, having either sorting property or blowing up to infinity in finite time.
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References
Bloch, A.M., Brockett, R.W., Ratiu, T.S.: Completely Integrable Gradient Flow. Commun. Math. Phys.147, 57–74 (1992)
Bogoyavlensky, O.I.: On perturbations of the Periodic Toda Lattice. Commun. Math. Phys.51, 201–209 (1976)
Cahn, R.N.: Semi-simple Lie Algebras and their Representations. Reading, MA: Benjamin/ Cummings, 1994
Chu, M.T., Norris, L.K.: Isospectral Flows and Abstract Matrix Factorizations. SIAM J. Numer. Anal.25, 1383–1391 (1988)
Deift, P., Li, L.C., Nanda, T., Tomei, C.: The Toda Flow on a Generic Orbit is Integrable. Comm. Pure Apple. Math.39, 183–232 (1986)
Deift, P., Li, L.C., Tomei, C.: Matrix factorizations and integrable systems. Comm. Pure Apple. Math.42, 443–521 (1989)
Ercolani, N., Flaschka, H., Singer, S.: The Geometry of the Full Kostant-Toda. Progress in Mathematics115, 181–226 (1993)
Flaschka, H.: On the Toda Lattice II. Prog. Theor. Phys.51, 703–716 (1974)
Gekhtman, M.I., Shapiro, M.Z.: Completeness of Real Toda Flows and Totally Positive Matrices. Preprint 1994
Humphreys, J.: Introduction to Lie Algebras and their Representations. New York: Springer, 1972
Kodama, Y., McLaughlin, K.: Explicit Integration of the Full Symmetric Toda hierarchy and the Sorting property. To appear in Lett. Math. Phys. (solv-int/9502006)
Kodama, Y., Ye, J.: Toda Hierarchy with Indefinite metric. To appear in Physica D. (solv-int /9505010)
Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. in Math.34, 195–338 (1979)
Manakov, S.V.: Complete Integrability and Stochastization of Discrete Dynamical Systems. Sov. Phys. JETP40, 703–716 (1975) (Zh. Exp. Teor. Fiz.,67, 543–555 (1974)
Moser, J.: Finitely Many Mass Points on the Line Under, the Influence of an Exponential Potential — An Integrable System. In: Dynamical Systems, Theory and Applications, J. Moser (ed.), Lect. Notes in Phys., Vol.38, Berlin and New York: Springer-Verlag, 1975, pp. 467–497.
Nanda, T.R.: Isospectral Flows on Banded Matrices. Ph.D. Dissertation, New York University, 1982
Symes, W.W.: The QR Algorithm and Scattering for the Finite Non-periodic Toda Lattice. Physica D4, 275–278 (1982)
Szegö, G.: Orthogonal Polynomials. AMS Colloquium Publications, Vol.23, 1939
Watkins, D.S.: Isospectral flows. SIAM Rev.26, No. 3, 379–392 (1984)
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Communicated by M. Jimbo
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Kodama, Y., Ye, J. Iso-spectral deformations of general matrix and their reductions on Lie algebras. Commun.Math. Phys. 178, 765–788 (1996). https://doi.org/10.1007/BF02108824
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DOI: https://doi.org/10.1007/BF02108824