Geodesic
Integrable deformations in the algebra of pseudodifferential
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 154-176
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We split the algebra of pseudodifferential operators in two different ways into the direct sum of two Lie subalgebras and deform the set of commuting elements in one subalgebra in the direction of the other component. The evolution of these deformed elements leads to two compatible systems of Lax equations that both have a minimal realization. We show that this Lax form is equivalent to a set of zero-curvature relations. We conclude by presenting linearizations of these systems, which form the key framework for constructing the solutions.
Keywords: integrable deformation, pseudodifferential operator, Kadomtsev–Petviashvili hierarchy, zero-curvature relation, linearization.
Mots-clés : Lax equation 
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G. F. Helminck; A. G. Helminck; E. A. Panasenko. Integrable deformations in the algebra of pseudodifferential. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 154-176. http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a10/

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

[2] G. Segal, G. Wilson, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 5–65 | DOI | MR | Zbl

[3] G. F. Helminck, A. G. Helminck, E. A. Panasenko, Solutions of the Strict KP Hierarchy, in preparation

[4] M. A. Olshanetsky, A. M. Perelomov, Phys. Rep., 71:5 (1981), 313–400 | DOI | MR

[5] M. Guest, Harmonic Maps, Loop Groups and Integrable Systems, London Mathematical Society Student Texts, 38, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[6] B. Kostant, Adv. Math., 34:3 (1979), 195–338 | DOI | MR | Zbl

[7] P. D. Lax, Commun. Pure Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

[8] M. Mulase, Adv. Math., 54:1 (1984), 57–66 | DOI | MR | Zbl