This article is cited in 18 papers

Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective

G. F. Helminck^a, A. G. Helminck^b, E. A. Panasenko<sup>c

a</sup> Korteweg-de~Vries Institute, University of Amsterdam, Amsterdam, The Netherlands
^b North Carolina State University, Raleigh, USA
^c Derzhavin Tambov State University, Tambov, Russia

Abstract: We split the algebra of pseudodifferential operators in two different ways into the direct sum of two Lie subalgebras and deform the set of commuting elements in one subalgebra in the direction of the other component. The evolution of these deformed elements leads to two compatible systems of Lax equations that both have a minimal realization. We show that this Lax form is equivalent to a set of zero-curvature relations. We conclude by presenting linearizations of these systems, which form the key framework for constructing the solutions.

Keywords: integrable deformation, pseudodifferential operator, Lax equation, Kadomtsev–Petviashvili hierarchy, zero-curvature relation, linearization.

Received: 14.05.2012

DOI: 10.4213/tmf8362

 English version: Theoretical and Mathematical Physics, 2013, 174:1, 134–153

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