On a ring of formal pseudodifferential operators.

*(English. Russian original)*Zbl 1008.37042
Proc. Steklov Inst. Math. 224, 266-280 (1999); translation from Tr. Mat. Inst. Steklova 224, 291-305 (1999).

A formal pseudodifferential operator in one variable is a formal series in powers of the operator on differentiation with respect to this variable such that the powers are bounded from above and coefficients are taken from some associative ring. The ring of such operators was introduced by Schur who considered the case when the ring of coefficients is commutative. It was revealed by Gel’fand and Dikii in the mid-1970s that this ring is useful in a formal calculus and soliton equations and, in particular, the Kadomtsev-Petviashvili hierarchy is described as a hierarchy of flows on such a ring.

In this paper the author considers the more general case of rings of formal pseudodifferential operators in several variables. The general algebraic theory of these rings is exposed. The author transfers to this theory many constructions known from the soliton theory and, in particular, introduces flows on such rings which are natural algebraic generalizations of the Kadomtsev-Petviashvili flows on the Schur ring. However, the relation of these flows to nonlinear differential equations is not discussed.

For the entire collection see [Zbl 0942.00074].

In this paper the author considers the more general case of rings of formal pseudodifferential operators in several variables. The general algebraic theory of these rings is exposed. The author transfers to this theory many constructions known from the soliton theory and, in particular, introduces flows on such rings which are natural algebraic generalizations of the Kadomtsev-Petviashvili flows on the Schur ring. However, the relation of these flows to nonlinear differential equations is not discussed.

For the entire collection see [Zbl 0942.00074].

Reviewer: Iskander A.Taimanov (Novosibirsk)

##### MSC:

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

16S32 | Rings of differential operators (associative algebraic aspects) |

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\textit{A. N. Parshin}, in: Algebra. Topologiya. Differentsial'nye uravneniya i ikh prilozheniya. Sbornik statej. K 90-letiyu so dnya rozhdeniya akademika L'va Semenovicha Pontryagina. Moskva: Nauka, Maik Nauka/ Interperiodika. 291--305 (1999; Zbl 1008.37042); translation from Tr. Mat. Inst. Steklova 224, 291--305 (1999)