Geodesic
On non-isospectral flows, Painlevé equations, and symmetries of differential and difference equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 93 (1992) no. 3, pp. 473-480 Cet article a éte moissonné depuis la source Math-Net.Ru

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We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called “discrete Painlevé equations” cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide “delay Painlevé equations”. 
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D. Levi; O. Ragnisco; M. A. Rodriguez. On non-isospectral flows, Painlevé equations, and symmetries of differential and difference equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 93 (1992) no. 3, pp. 473-480. http://geodesic.mathdoc.fr/item/TMF_1992_93_3_a5/

[1] Painlevé P., Acta Math., 25 (1902), 1–85 | DOI | MR

[2] Gambier G., Acta Math., 33 (1909), 1–55 | DOI | MR | Zbl

[3] Ablowitz M. J., Ramani A., Segur H., Lett. Nuovo Cimento, 23 (1978), 333–338 | DOI | MR

[4] Flaschka H., Newell A. C., Comm. Math. Phys., 76 (1980), 65 | DOI | MR | Zbl

[5] Jimbo M., Miwa T., Ueno K., Physica D, 2 (1981), 306 | DOI | MR | Zbl

[6] Its A. R., Novokshenov V. Yu., The isomonodromic deformation method in the theory of Painlevé equations, Lect. Notes in Math., 1191, Springer-Verlag, 1986 | DOI | MR | Zbl

[7] Fokas A. S., Zhou X., On the solvability of Painlevé II and IV, Preprint Claikson University, INS-148, january 1990 | MR

[8] Fokas A. S., Its A. R., Zhou X., Painlevé transcendents (Sainte-Adèle, PQ, 1990), NATO Adv. Sci. Inst. Ser., 278, Plenum, New York, 1992, 33–47 | MR | Zbl

[9] Papageorgiou V. G., Nijhoff F., Phys. Lett. A, 53 (1991), 337 | MR

[10] Papageorgiou V. G., Grammaticos B., Nijhoff F., Ramani A., Isomonodromic Deformation Problems for Discrete Analogues of Painlevé equations, preprint INS-178/91 | MR

[11] Fuchssteiner B., Progr. Theor. Phys., 70 (1983), 1508–1522 ; Oevel W., Mastersymmetries: weak action-angle structure for hamiltonian and non-hamiltonian dynamical systems, preprint, Paderborn, 1986 | DOI | MR | Zbl | MR | Zbl

[12] Levi D., Ragnisco O., Lett. Nuovo Cimento, 22 (1978), 691 ; Bruschi M., Levi D., Ragnisco O., II Nuovo Cimento A, 48 (1978), 213–226 ; Levi D., Ragnisco O., J. Phys. A; Math. Gen., 7 (1979), L157–L162 | DOI | MR | DOI | MR | DOI | MR