Abstract
We show that the analog of the Miura maps and Bäcklund-Darboux transformations for a general class of equations of Toda type and for a generalized class of periodic Toda flows are isomorphisms of Poisson Lie groups.
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[A] Adler, M.: On the Bäcklund transformation for the Gelfand-Dickey equations. Commun. Math. Phys.80, 517–527 (1981)
[C] Chu, M.: A differential equation approach to the singular value decomposition of bidiagonal matrices. Lin. Alg. Appl.80, 71–80 (1986)
[D] Deift, P.: Applications of a commutation formula. Duke Math. J.45, 267–310 (1978)
[DDLT1] Deift, P., Demmel, J., Li, L.-C., Tomei, C.: The bidiagonal singular value decomposition and Hamiltonian mechanics. SIAM. J. Numer. Anal.28, 1463–1516 (1991)
[Drin] Drinfeld, V.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Sov. Math. Dokl.27, 68–71 (1983)
[F] Flaschka, H.: Private communication
[GD1] Gelfand, I., Dickey, L.: Fractional power of operators and Hamiltonian systems. Funct. Anal. Appl.10, 259–273 (1976)
[GD2] Gelfand, I., Dickey, L.: A family of Hamiltonian structures related to nonlinear integrable differential equations. Preprint no. 136, Inst. Appl. Math. USSR Acad. Sci. 1978 (in Russian), English transl. in Collected papers of I. M. Gelfand, Vol. 1, Berlin, Heidelberg, New York: Springer 1987, pp. 625–646
[GW] Goodman, R., Wallach, N.: Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math.347, 69–133 (1984)
[H] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York; Academic Press 1978
[KW] Kupershmidt, B., Wilson, G.: Modifying Lax equations and second Hamiltonian structure. Invent. Math.62, 403–436 (1981)
[KvM] Kac, M., van Moerbeke, P.: On some perodic Toda lattices. Proc. Natl. Acad. Sci. USA72, 1627–1629 (1975)
[LP] Li, L.-C., Parmentier, S.: A new class of quadratic Poisson structures and the Yang-Baxter equation. C. R. Acad. Sci., Paris Sér. I. Math.307, 279–281 (1988). Nonlinear Poisson structures andr-matrices. Commun. Math. Phys.125, 545–563 (1989)
[M] Miura, R.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys.9, 1202–1204 (1968)
[STS1] Semenov-Tian-Shansky, M.: What is a classicalr-matrix? Funct. Anal. Appl.17, 259–272 (1983)
[STS2] Semonov-Tian-Shansky, M.: Dressing transformations and Poisson group actions. Publ. RIMS, Kyoto Univ.21, 1237–1260 (1985)
[V] Varadarajan, V.S.: Lie groups, Lie algebras, and their representations. Berlin, Heidelberg, New York: Springer 1984
[WE] Wahlquist, H., Estabrook, F.: Bäcklund transformation for solutions of the Kortewegde Vries equation. Phys. Rev. Lett.31, 1386–1390 (1973)
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Deift, P., Li, LC. Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows. Commun.Math. Phys. 143, 201–214 (1991). https://doi.org/10.1007/BF02100291
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DOI: https://doi.org/10.1007/BF02100291