Geodesic
Properties of the algebra psd related to integrable hierarchies
Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 130, pp. 183-195
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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, n-KdV hierarchy, KP hierarchy, strict KP hierarchy
Mots-clés : constant term, Lax equations. 
@article{VTAMU_2020_25_130_a6,
     author = {G. F. Helminck and E. A. Panasenko},
     title = {Properties of the algebra psd related to integrable hierarchies},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {183--195},
     year = {2020},
     volume = {25},
     number = {130},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2020_25_130_a6/}
}
TY  - JOUR
AU  - G. F. Helminck
AU  - E. A. Panasenko
TI  - Properties of the algebra psd related to integrable hierarchies
JO  - Vestnik rossijskih universitetov. Matematika
PY  - 2020
SP  - 183
EP  - 195
VL  - 25
IS  - 130
UR  - http://geodesic.mathdoc.fr/item/VTAMU_2020_25_130_a6/
LA  - ru
ID  - VTAMU_2020_25_130_a6
ER  - 
%0 Journal Article
%A G. F. Helminck
%A E. A. Panasenko
%T Properties of the algebra psd related to integrable hierarchies
%J Vestnik rossijskih universitetov. Matematika
%D 2020
%P 183-195
%V 25
%N 130
%U http://geodesic.mathdoc.fr/item/VTAMU_2020_25_130_a6/
%G ru
%F VTAMU_2020_25_130_a6
G. F. Helminck; E. A. Panasenko. Properties of the algebra psd related to integrable hierarchies. Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 130, pp. 183-195. http://geodesic.mathdoc.fr/item/VTAMU_2020_25_130_a6/

[1] P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math., 21:5 (1968), 467–490 | DOI | MR | Zbl

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

[3] I. M. Gelfand, L. A. Dickey, “Fractional powers of operators and Hamiltonian systems”, Funct. Anal. Its Appl., 10:4 (1976), 259–273 | DOI | MR

[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 | DOI | MR

[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), World Sci. Publishing, Singapore, 1983, 39–119 | MR

[6] G. Segal, G. Wilson, “Loop groups and equations of KdV type”, Publications Mathematiques de l'IHES, 61 (1985), 5–65 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[10] G. F. Helminck, E. A. Panasenko, “Geometric solutions of the strict KP hierarchy”, Theoret. and Math. Phys., 198:3 (2019), 48–68 | DOI | MR | Zbl

[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 | DOI | MR | Zbl