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Properties of the algebra Psd related to integrable hierarchies

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In this paper we discuss and prove various properties of the algebra of pseudo differential operators related to integrable hierarchies in this algebra, in particular the KP hierarchy and its strict version. Some explain the form of the equations involved or give insight in why certain equations in these systems are combined, others lead to additional properties of these systems like a characterization of the eigenfunctions of the linearizations of the mentioned hierarchies, the description of elementary Darboux transformations of both hierarchies and the search for expressions in Fredholm determinants for the constructed eigenfunctions and their duals.

Keywords

pseudo differential operators; the adjoint; constant term; n -KdV hierarchy; KP hierarchy; strict KP hierarchy; Lax equations

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DOI

10.20310/2686-9667-2020-25-130-183-195

UDC

517.958+517.98

Pages

183-195

References

[1] P.D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math., 21:5 (1968), 467–490. [2] G. Wilson, “Commuting flows and conservation laws for Lax equations”, Math. Proc. Camb. Phil. Soc., 86:1 (1979), 131–143. [3] I. M. Gelfand, L. A. Dickey, “Fractional powers of operators and Hamiltonian systems”, Funct. Anal. Its Appl., 10:4 (1976), 259–273. [4] M. Sato, Y. Sato, “Soliton equations as dynamical systems on infinite-dimensional Grassman manifold”, Nonlinear Partial Differential Equations In Applied Science, Proceedings of the U.S.-Japan seminar “Nonlinear partial differential equations in applied science” (Tokyo, 1982 (North-Holland mathematics studies)), 1983, 259–272. [5] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Transformation groups for soliton equations”, Non-Linear Integrable Systems–Classical Theory and Quantum Theory, Proceedings of RIMS symposium “Non-linear integrable systems–classical theory and quantum theory” (Kyoto, Japan, 13-16 May, 1981), 1983, 39–119. [6] G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publications Mathematiques de l’IHES, 61 (1985), 5–65. [7] G.F. Helminck, A.G. Helminck, E.A. Panasenko, “Integrable deformations in the algebra of pseudo differential operators from a Lie algebraic perspective”, Theoret. and Math. Phys., 174:1 (2013), 134–153. [8] G.F. Helminck, E.A. Panasenko, S.V. Polenkova, “Bilinear equations for the strict KP hierarchy”, Theoret. and Math. Phys., 185:3 (2015), 1804–1816. [9] G.F. Helminck, A.G. Helminck, E.A. Panasenko, “Cauchy problems related to integrable deformations of pseudo differential operators”, Journal of Geometry and Physics, 85 (2014), 196–205. [10] G.F. Helminck, E.A. Panasenko, “Geometric solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 198:3 (2019), 48–68. [11] G.F. Helminck, E.A. Panasenko, “Expressions in Fredholm determinants for solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 199:2 (2019), 637–651.

Received

2020-03-24

Section of issue

Scientific articles

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