Strict versions of integrable hierarchies in pseudodifference operators and the related Cauchy problems

G. F. Helminck ; V. A. Poberezhny ; S. V. Polenkova

Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 2, pp. 225-245

Résumé

In the algebra $Ps\Delta$ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator $\Lambda_0$. The first deformation is by the group in $Ps\Delta$ corresponding to the Lie subalgebra $Ps\Delta_{<0}$ of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra $Ps\Delta_{\le0}$ of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on $\Lambda_0$ that respectively complements the Lie subalgebra $Ps\Delta_{<0}$ or $Ps\Delta_{\le0}$. This yields two integrable hierarchies associated with $\Lambda_0$, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For $\Lambda_0$ equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.

Détail

Keywords: pseudodifference operator, zero-curvature form, Cauchy problem.
Mots-clés : Lax equation

Bibliographie

[1] E. Date, M. Kashiwara, M. Jimbo, T. Miwa, “Transformation groups for soliton equations”, Nonlinear Integrable Systems – Classical Theory and Quantum Theory (Kyoto, Japan 13–16 May, 1981), eds. M. Jimbo, T. Miwa, World Sci., Singapore, 1983, 41–119 | MR

[2] G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Inst. Hautes Étud. Sci. Publ. Math., 61 (1985), 5–65 | MR | MR | Zbl

[3] G. F. Khelmink, A. G. Khelmink, E. A. Panasenko, “Integriruemye deformatsii v algebre psevdodifferentsialnykh operatorov s tochki zreniya algebraicheskoi teorii Li”, TMF, 174:1 (2013), 154–176 | DOI | MR | Zbl

[4] M. Toda, Nonlinear Waves and Solitons, Mathematics and Its Applications (Japanese Series), 5, Kluwer, Dordrecht; KTK Scientific Publishers, Tokyo, 1989 | MR

[5] K. Ueno, K. Takasaki, “Toda lattice hierarchy”, Group Representations and Systems of Differential Equations (University of Tokyo, December 20–27, 1982), Advanced Studies in Pure Mathematics, 4, ed. K. Okamoto, North-Holland, Amsterdam–New York, 1984, 1–95 | MR

[6] T. Takebe, “Toda lattice hierarchy and conservation laws”, Commun. Math. Phys., 129:2 (1990), 281–318 | DOI | MR

[7] M. Adler, P. van Moerbeke, “Group factorization, moment matrices, and Toda lattices”, Internat. Math. Res. Notices, 1997:12 (1997), 555–572 | DOI | MR

[8] L. Haine, P. Iliev, “Commutative rings of difference operators and an adelic flag manifold”, Internat. Math. Res. Notices, 2000:6 (2000), 281–323 | DOI | MR

[9] D. Olive, N. Turok, “Algebraic structure of Toda systems”, Nucl. Phys. B, 220:4 (1983), 491–507 | DOI | MR

[10] D. Olive, N. Turok, “Local conserved densities and zero-curvature conditions for Toda lattice field theories”, Nucl. Phys. B, 257:2 (1985), 277–301 | DOI | MR

[11] D. Olive, N. Turok, “The Toda lattice field theory hierarchies and zero-curvature conditions in Kac–Moody algebras”, Nucl. Phys. B, 265:3 (1986), 469–484 | DOI | MR

[12] M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic Integrability, Painlevé Geometry and Lie Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, 47, Springer, Berlin | DOI | MR

[13] M. Adler, P. van Moerbeke, “Vertex operator solutions to the discrete KP-hierarchy”, Commun. Math. Phys., 203:1 (1999), 185–210 | DOI | MR

[14] L. A. Dickey, Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, 26, World Sci., Singapore, 2003 | DOI | MR

[15] G. Wilson, “Commuting flows and conservation laws for Lax equations”, Math. Proc. Cambridge Philos. Soc., 86:1 (1979), 131–143 | DOI | MR

[16] M. Adler, E. Horozov, P. van Moerbeke, “The solution to the $q$-KdV equation”, Phys. Lett. A, 242:3 (1998), 139–151 | DOI | MR

[17] R. R. Gontsov, V. A. Poberezhnyi, G. F. Khelmink, “Deformatsii sistem lineinykh differentsialnykh uravnenii”, UMN, 66:1(397) (2011), 65–110 | DOI | DOI | MR | Zbl

[18] G. F. Helminck, J. W. van de Leur, “Darboux transformations for the KP hierarchy in the Segal–Wilson setting”, Canad. J. Math., 53:2 (2001), 278–309 | DOI | MR

[19] G. F. Helminck, V. A. Poberezhny, S. V. Polenkova, “A geometric construction of solutions of the strict $\text{dKP}(\Lambda_0$) hierarchy”, J. Geom. Phys., 131 (2018), 189–203 | DOI | MR