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Poisson-Lie group of pseudodifferential symbols and fractional KP-KdV hierarchies. (English. Abridged French version) Zbl 0771.35056
Summary: The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the “logarithmic” 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the Lie group of classical symbols of all real (or complex) degrees. It turns out that this group has a natural Poisson-Lie structure whose restriction to differential operators of an arbitrary integer order coincides with the second Adler-Gelfand-Dickey structure. Moreover, for any real (or complex) \(\alpha\) there exists a hierarchy of completely integrable equations on the degree \(\alpha\) pseudodifferential symbols, and this hierarchy for \(\alpha=1\) coincides with the KP one, and for an integer \(\alpha=n\geq 2\) and purely differential symbol gives the \(n\)-KdV-hierarchy.

35Q53 KdV equations (Korteweg-de Vries equations)
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J70 Invariance and symmetry properties for PDEs on manifolds
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