mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2020-25-130-183-195mathematics-219Research ArticleСТАТЬИARTICLESСвойства алгебры псевдодифференциальных операторов, связанные с интегрируемыми иерархиямиProperties of the algebra Psd related to integrable hierarchiesХельминкГерард ФранцискHelminckGerard F.g.f.helminck@uv.nlПанасенкоЕлена АлександровнаPanasenkoElena A.panlena_t@mail.ruМатематический институт Кортевега - де Фриза, Университет г. АмстердамРоссияKdV Institute, University of AmsterdamRussian FederationФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина»РоссияDerzhavin Tambov State UniversityRussian Federation20201607202025130183195Copyright © Хельминк Г.Ф., Панасенко Е.А., 20202020Хельминк Г.Ф., Панасенко Е.А.Helminck G.F., Panasenko E.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/219

В работе рассматриваются различные свойства алгебры псевдодифференциальных операторов, связанные с интегрируемыми иерархиями, возникающими в этой алгебре, в частности, иерархией Кадомцева-Петвиашвили (КП) и ее строгой версией. Одни свойства проясняют вид уравнений в иерархиях и дают понимание того, почему уравнения определенного вида скомбинированы в этих системах, другие позволяют изучить свойства самих систем, а именно: вид собственных функций линеаризаций упомянутых иерархий, описание элементарных преобразований Дарбу обоих иерархий, отыскание представлений построенных собственных функций и двойственных им в терминах определителей Фредгольма.

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.

псевдодифференциальные операторысопряженный операторсвободный члениерархия n-КдФиерархия КПстрогая иерархия КПуравнения Лаксаpseudo differential operatorsthe adjointconstant termn -KdV hierarchyKP hierarchystrict KP hierarchyLax equationsReferencesP.D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math., 21:5 (1968), 467-490.P.D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math., 21:5 (1968), 467-490.G. Wilson, “Commuting flows and conservation laws for Lax equations”, Math. Proc. Camb. Phil. Soc., 86:1 (1979), 131-143.G. Wilson, “Commuting flows and conservation laws for Lax equations”, Math. Proc. Camb. Phil. Soc., 86:1 (1979), 131-143.I. M. Gelfand, L. A. Dickey, “Fractional powers of operators and Hamiltonian systems”, Funct. Anal. Its Appl., 10:4 (1976), 259-273.I. M. Gelfand, L. A. Dickey, “Fractional powers of operators and Hamiltonian systems”, Funct. Anal. Its Appl., 10:4 (1976), 259-273.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.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.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.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.G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publications Mathematiques de l’IHES, 61 (1985), 5-65.G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publications Mathematiques de l’IHES, 61 (1985), 5-65.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.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.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.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.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.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.G.F. Helminck, E.A. Panasenko, “Geometric solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 198:3 (2019), 48-68.G.F. Helminck, E.A. Panasenko, “Geometric solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 198:3 (2019), 48-68.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.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.

The authors declare that there are no conflicts of interest present.