Geodesic
Integrable symplectic maps via reduction of Bäcklund transformation
Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 1, pp. 39-50 Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss the stationary potential equations as illustrative examples to explain how to construct integrable symplectic maps via Bäcklund transformations. We first give a terse survey of Bäcklund transformations of the potential KdV equation and the potential fifth-order KdV equation. Then, using Jacobi–Ostrogradsky coordinates, we obtain canonical Hamiltonian forms of the stationary potential equations. Finally, we construct symplectic maps from the reduction of a Bäcklund transformation and verify that they are integrable.
Keywords: integrable symplectic map, stationary potential KdV equation, Bäcklund transformation, Lax representation. 
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Dianlou Du; Yuanyuan Lui; Xue Wang. Integrable symplectic maps via reduction of Bäcklund transformation. Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 1, pp. 39-50. http://geodesic.mathdoc.fr/item/TMF_2021_208_1_a2/

[1] J. Hietarinta, N. Joshi, F. W. Nijhoff, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, 54, Cambridge Univ. Press, Cambridge, 2016 | DOI | MR

[2] V. E. Adler, A. I. Bobenko, Yu. B. Suris, “Classification of integrable equations on quad- graphs. The consistency approach”, Commun. Math. Phys., 233:3 (2003), 513–543 | DOI | MR

[3] M. Adler, P. van Moerbeke, “Toda–Darboux maps and vertex operators”, Internat. Math. Res. Notices, 1998, no. 10, 489–511 | DOI | MR

[4] L. D. Faddeev, A. Y. Volkov, “Hirota equation as an example of integrable symplectic map”, Lett. Math. Phys., 32:2 (1994), 125–135 | DOI | MR

[5] B. Grammaticos, T. Tamizhmani, Y. Kosmann-Schwarzbach (eds.), Discrete Integrable Systems, Lecture Notes in Physics, 644, Springer, Berlin, Heidelberg, Berlin, 2004 | DOI

[6] J. Moser, A. P. Veselov, “Discrete versions of some classical integrable systems and factorization of matrix polynomials”, Commun. Math. Phys., 139:2 (1991), 217–243 | DOI | MR

[7] M. Bruschi, O. Ragnisco, P. M. Santini, G. Z. Tu, “Integrable symplectic maps”, Phys. D, 49:3 (1991), 273–294 | DOI | MR

[8] Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219, Birkhäuser, Basel, 2003 | DOI | MR

[9] A. P. Veselov, “Integriruemye otobrazheniya”, UMN, 46:5(281) (1991), 3–45 | DOI | MR | Zbl

[10] D. Du, “Complex form, reduction and Lie–Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy”, Phys. A, 303:3–4 (2002), 439–456 | DOI | MR

[11] D. Du, C. Cao, “The Lie–Poisson representation of the nonlinearized eigenvalue problem of the Kac–van Moerbeke hierarchy”, Phys. Lett. A, 278:4 (2001), 209–224 | DOI | MR

[12] Y. Wu, D. Du, “On the Lie–Poisson structure of the nonlinearized discrete eigenvalue problem”, J. Math. Phys., 41:8 (2000), 5832–5848 | DOI | MR

[13] C. Cao, X. Xu, “A finite genus solution of the H1 model”, J. Phys. A: Math. Theor., 45:5 (2012), 055213, 13 pp. | DOI | MR

[14] C. Cao, G. Zhang, “A finite genus solution of the Hirota equation via integrable symplectic maps”, J. Phys. A: Math. Theor., 45:9 (2012), 095203, 25 pp. | DOI | MR

[15] C. Cao, Y. Wu, X. Geng, “Relation between the Kadomtsev–Petviashvili equation and the confocal involutive system”, J. Math. Phys., 40:8 (1999), 3948–3970 | DOI | MR

[16] C. Cao, “Nonlinearization of the Lax system for AKNS hierarchy”, Sci. China Ser. A, 33:5 (1990), 528–536 | MR

[17] C. Cao, X. Geng, “C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy”, J. Phys. A: Math. Gen., 23:18 (1990), 4117–4125 | DOI | MR

[18] Y. Wu, X. Geng, “A new integrable symplectic map of Neumann type”, J. Phys. Soc. Japan, 68:3 (1999), 784–790 | DOI | MR

[19] A. P. Fordy, “Integrable symplectic maps”, Symmetries and Integrability of Difference Equations, London Mathematical Society Lecture Note Series, 225, eds. P. A. Clarkson, F. W. Nijhoff, Cambridge Univ. Press, Cambridge, 1999, 43–55 | DOI | MR

[20] A. P. Fordi, A. B. Shabat, A. P. Veselov, “Faktorizatsiya i puassonovy sootvetstviya”, TMF, 105:2 (1995), 225–245 | DOI | MR | Zbl

[21] Dzh. Lemb, Elementy teorii solitonov, Mir, M., 1984 | MR