Pogrebkov A.K. Negative numbers of times of integrable hierarchies (abstract)

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Speaker: Andrei Pogrebkov

Title: Negative numbers of times of integrable hierarchies

Abstract:
Time evolutions of the dressing operators of the integrable hierarchies, like Kadomtsev-Petviashvili or Davey-Stewartson, are given by linear differential operators. In the standard situation these operators result from dressing of positive powers of a [math]\displaystyle{ \partial_x }[/math]. It is natural to call these semiinfinite hierarchies of integrable (2+1)-dimensional equations as hierarchies with positive numbers of times. Here we develop hierarchies directed to negative numbers of times. Derivation of such systems, as well as of the corresponding hierarchies, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lift up to nonlinear integrable ones by means of the special dressing procedure. Thus one can construct not only nonlinear equations but corresponding Lax pairs, as well. Lax operator of such evolutions coincide with the Lax operator of the "positive" hierarchy. We also derive (1+1)-dimensional reductions of equations of such hierarchies.


Event: Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.
Alexandre Vinogradov Memorial Conference.