Geodesic
Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy
Symmetry, integrability and geometry: methods and applications, Tome 2 (2006) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlevé equation $\mathrm{dP}_{\rm I}$ in a sense that autonomous version of these equations admit the limit to the first Painlevé equation. It is shown that each of these equations describes Bäcklund transformations of Veselov–Shabat periodic dressing lattices with odd period known also as Noumi–Yamada systems of type $A_{2(n-1)}^{(1)}$.
Keywords: extended discrete KP hierarchy; similarity reductions; discrete Painlevé equations. 
@article{SIGMA_2006_2_a4,
     author = {Andrei K. Svinin},
     title = {Integrable {Discrete} {Equations} {Derived} by {Similarity} {Reduction} of the {Extended} {Discrete} {KP} {Hierarchy}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2006},
     volume = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a4/}
}
TY  - JOUR
AU  - Andrei K. Svinin
TI  - Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2006
VL  - 2
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a4/
LA  - en
ID  - SIGMA_2006_2_a4
ER  - 
%0 Journal Article
%A Andrei K. Svinin
%T Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy
%J Symmetry, integrability and geometry: methods and applications
%D 2006
%V 2
%U http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a4/
%G en
%F SIGMA_2006_2_a4
Andrei K. Svinin. Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy. Symmetry, integrability and geometry: methods and applications, Tome 2 (2006). http://geodesic.mathdoc.fr/item/SIGMA_2006_2_a4/

[1] Adler V. E., “Nonlinear chains and Painlevé equations”, Phys. D, 73 (1994), 335–351 | DOI | MR | Zbl

[2] Airault H., “Rational solutions of Painlevé equations”, Stud. Appl. Math., 61 (1979), 31–53 | MR | Zbl

[3] Bogoyavlenskii O. I., Breaking solitons: nonlinear integrable equations, Nauka, Moscow, 1991 (in Russian) | MR | Zbl

[4] Casati P., Falqui G., Magri F., Pedroni, M., The KP theory revisited I, II, III, IV, SISSA Preprint, SISSA/2-5/96/FM, 1996

[5] Cresswell C., Joshi N., “The discrete first, second and thirty-fourth Painlevé hierarchies”, J. Phys. A: Math. Gen., 32 (1999), 655–669 | DOI | MR | Zbl

[6] Flaschka H., Newell A. C., “Monodromy- and spectrum-preserving deformations, I”, Comm. Math. Phys., 76 (1980), 65–116 | DOI | MR | Zbl

[7] Fokas A. S., Its A. R., Kitaev A. R., “The isomonodromy approach to matrix models in 2D quantum gravity”, Comm. Math. Phys., 147 (1992), 395–430 | DOI | MR | Zbl

[8] Fokas A. S., Grammaticos B., Ramani A., “From continuous to discrete Painlevé equations”, J. Math. Anal. Appl., 180 (1993), 342–360 | DOI | MR | Zbl

[9] Grammaticos B., Ramani A., Papageorgiou V., “Do integrable mapping have the Painlevé property?”, Phys. Rev. Lett., 67 (1991), 1825–1828 | DOI | MR | Zbl

[10] Grammaticos B., Papageorgiou V., Ramani A., “Discrete dressing transformations and Painlevé equations”, Phys. Lett. A, 235 (1997), 475–479 | DOI | MR

[11] Grammaticos B., Ramani A., Satsuma J., Willox R., Carstea A. S., “Reductions of integrable lattices”, J. Nonlinear Math. Phys., 12, suppl. 1 (2005), 363–371 | DOI | MR

[12] Kazakova T. G., “Finite-dimensional reductions of the discrete Toda chain”, J. Phys. A: Math. Gen., 37 (2004), 8089–8102 | DOI | MR | Zbl

[13] Magri F., Pedroni M., Zubelli J. P., “On the geometry of Darboux transformations for the KP hierarchy and its connection with the discrete KP hierarchy”, Comm. Math. Phys., 188 (1997), 305–325 | DOI | MR | Zbl

[14] Nijhoff F. W., Satsuma J., Kajiwara K., Grammaticos B., Ramani A., “A study of the alternative discrete Painlevé-II equation”, Inverse Problems, 12 (1996), 697–716 | DOI | MR | Zbl

[15] Nijhoff F. W., Papageorgiou V. G., “Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation”, Phys. Lett. A, 153 (1991), 337–344 | DOI | MR

[16] Noumi M., Yamada Y., “Affine Weyl groups, discrete dynamical systems and Painlevé equations”, Comm. Math. Phys., 199 (1998), 281–295 | DOI | MR | Zbl

[17] Noumi M., Yamada Y., “Higher order Painlevé equations of type $A_l^{(1)}$”, Funkcial. Ekvac., 41 (1998), 483–503 | MR | Zbl

[18] Svinin A. K., “Extension of the discrete KP hierarchy”, J. Phys. A: Math. Gen., 35 (2002), 2045–2056 | DOI | MR | Zbl

[19] Svinin A. K., “Extended discrete KP hierarchy and its reductions from a geometric viewpoint”, Lett. Math. Phys., 61 (2002), 231–239 | DOI | MR | Zbl

[20] Svinin A. K., “Invariant submanifolds of the Darboux–Kadomtsev–Petviashvili chain and an extension of the discrete Kadomtsev–Petviashili hierarchy”, Theor. Math. Phys., 141:2 (2004), 1542–1561 | DOI | MR | Zbl

[21] Ueno K., Takasaki K., “Toda lattice hierarchy. I, II”, Proc. Japan Acad. Ser. A Math. Sci., 59 (1983), 167–170 ; 215–218 | DOI | MR | Zbl | MR | Zbl

[22] Veselov A. P., Shabat A. B., “Dressing chains and the spectral theory of the Schrödinger operator”, Funct. Anal. Appl., 27:2 (1993), 81–96 | DOI | MR | Zbl