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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
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.
Received: 25.12.2020 Revised: 25.12.2020
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
Linking options:
https://www.mathnet.ru/eng/tmf10046https://doi.org/10.4213/tmf10046 https://www.mathnet.ru/eng/tmf/v207/i3/p458
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