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Teoreticheskaya i Matematicheskaya Fizika, 2021, Volume 207, Number 3, Pages 458–488
DOI: https://doi.org/10.4213/tmf10046
(Mi tmf10046)
 

This article is cited in 3 scientific papers (total in 3 papers)

On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators

J.-P. Magnota, V. N. Rubtsovbcd

a Lycée Jeanne d'Arc, Clermont-Ferrand, France
b Laboratoire Angevin de Recherche en Mathématiques, Université d’Angers, Angers, France
c Institute for Theoretical and Experimental Physics, Moscow, Russia
d Institute for Geometry and Physics, Trieste, Italy
Full-text PDF (670 kB) Citations (3)
References:
Abstract: We study the existence and uniqueness of the Kadomtsev–Petviashvili (KP) hierarchy solutions in the algebra $\mathcal FCl(S^1,\mathbb K^n)$ of formal classical pseudodifferential operators. The classical algebra $\Psi DO(S^1,\mathbb K^n)$, where the KP hierarchy is well known, appears as a subalgebra of $\mathcal FCl(S^1,\mathbb K^n)$. We investigate algebraic properties of $\mathcal FCl(S^1,\mathbb K^n)$ such as splittings, $r$-matrices, extension of the Gelfand–Dickey bracket, and almost complex structures. We then prove the existence and uniqueness of the KP hierarchy solutions in $\mathcal FCl(S^1,\mathbb K^n)$ with respect to extended classes of initial values. Finally, we extend this KP hierarchy to complex-order formal pseudodifferential operators and describe their Hamiltonian structures similarly to the previously known formal case.
Keywords: formal pseudodifferential operator, Kadomtsev–Petviashvili hierarchy, almost complex structure, almost quaternionic structure.
Funding agency Grant number
Russian Foundation for Basic Research 18-01-00461
Centre National de la Recherche Scientifique LAREMA UMR 6093
IPaDEGAN 778010
This research of both authors was supported by LAREMA UMR 6093 du CNRS. V.R. was partly supported by the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010, and by the Russian Foundation for Basic Research under the Grant RFBR 18-01-00461.
Received: 25.12.2020
Revised: 25.12.2020
English version:
Theoretical and Mathematical Physics, 2021, Volume 207, Issue 3, Pages 799–826
DOI: https://doi.org/10.1134/S004057792106009X
Bibliographic databases:
Document Type: Article
MSC: 37K10, 37K20, 37K30
Language: Russian
Citation: J.-P. Magnot, V. N. Rubtsov, “On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators”, TMF, 207:3 (2021), 458–488; Theoret. and Math. Phys., 207:3 (2021), 799–826
Citation in format AMSBIB
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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