Geodesic
Canonicity of Bäcklund Transformation: $r$-Matrix Approach. II
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical physics. Problems of quantum field theory, Tome 226 (1999), pp. 134-139
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This work represents the second part of the paper devoted to the general proof of the canonicity of the Bäcklund transformation (BT) for Hamiltonian integrable systems described by an $SL(2)$-invariant $r$-matrix. Introducing an extended phase space from which the original space is obtained by imposing first-kind constraints, one can prove the canonicity of the BT by a new method. This new proof provides a natural explanation for the fact why the gauge transformation of the matrix $M$ associated with the BT has the same structure as the Lax operator $L$. This technique is illustrated through an example of a DST chain. 
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     author = {E. K. Sklyanin},
     title = {Canonicity of {B\"acklund} {Transformation:} $r${-Matrix} {Approach.~II}},
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E. K. Sklyanin. Canonicity of Bäcklund Transformation: $r$-Matrix Approach. II. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical physics. Problems of quantum field theory, Tome 226 (1999), pp. 134-139. http://geodesic.mathdoc.fr/item/TM_1999_226_a9/

[1] Sklyanin E. K., “Canonicity of Bäcklund transformation: $r$-matrix approach, I”, AMS Collection dedicated to L. D. Faddeev seminar (to appear)

[2] Takhtadzhyan L. A., Faddeev L. D., Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[3] Kuznetsov V. B., Sklyanin E. K., “On Bäcklund transformations for many-body systems”, J. Phys. A: Math. Gen., 31 (1998), 2241–2251 | DOI | MR | Zbl

[4] Pasquier V., Gaudin M., “The periodic Toda chain and a matrix generalization of the Bessel function recursion relations”, J. Phys. A: Math. Gen., 25 (1992), 5243–5252 | DOI | MR | Zbl

[5] Baxter R. I., Exactly solved models in statistical mechanics, ch. 9–10, Acad. Press, London, 1982 | MR | Zbl