144 research outputs found
Constrained KP Hierarchies: Additional Symmetries, Darboux-B\"{a}cklund Solutions and Relations to Multi-Matrix Models
This paper provides a systematic description of the interplay between a
specific class of reductions denoted as \cKPrm () of the primary
continuum integrable system -- the Kadomtsev-Petviashvili ({\sf KP}) hierarchy
and discrete multi-matrix models. The relevant integrable \cKPrm structure is a
generalization of the familiar -reduction of the full {\sf KP} hierarchy to
the generalized KdV hierarchy . The important feature
of \cKPrm hierarchies is the presence of a discrete symmetry structure
generated by successive Darboux-B\"{a}cklund (DB) transformations. This
symmetry allows for expressing the relevant tau-functions as Wronskians within
a formalism which realizes the tau-functions as DB orbits of simple initial
solutions. In particular, it is shown that any DB orbit of a
defines a generalized 2-dimensional Toda lattice structure. Furthermore, we
consider the class of truncated {\sf KP} hierarchies ({\sl i.e.}, those defined
via Wilson-Sato dressing operator with a finite truncated pseudo-differential
series) and establish explicitly their close relationship with DB orbits of
\cKPrm hierarchies. This construction is relevant for finding partition
functions of the discrete multi-matrix models.
The next important step involves the reformulation of the familiar
non-isospectral additional symmetries of the full {\sf KP} hierarchy so that
their action on \cKPrm hierarchies becomes consistent with the constraints of
the reduction. Moreover, we show that the correct modified additional
symmetries are compatible with the discrete DB symmetry on the \cKPrm DB
orbits.
The above technical arsenal is subsequently applied to obtain completeComment: LaTeX, 63 pg
Supersymmetric KP Hierarchy: ``Ghost'' Symmetry Structure, Reductions and Darboux-Backlund Solutions
This paper studies Manin-Radul supersymmetric KP hierarchy (MR-SKP) in three
related aspects: (i) We find an infinite set of additional (``ghost'') symmetry
flows spanning the same (anti-)commutation algebra as the ordinary MR-SKP
flows; (ii) The latter are used to construct consistent reductions of the
initial unconstrained MR-SKP hierarchy which involves a nontrivial modification
for the fermionic flows; (iii) For the simplest constrained MR-SKP hierarchy we
show that the orbit of Darboux-Backlund transformations lies on a
supersymmetric Toda lattice being a square-root of the standard one-dimensional
Toda lattice, and also we find explicit Wronskian-ratio solutions for the
super-tau function.Comment: Minor corrections in few equations. LaTeX, 12 pg
R-Matrix Formulation of KP Hierarchies and their Gauge Equivalence
The Adler-Kostant-Symes -bracket scheme is applied to the algebra of
pseudo-differential operators to relate the three integrable hierarchies: KP
and its two modifications, known as nonstandard integrable models. All three
hierarchies are shown to be equivalent and connection is established in the
form of a symplectic gauge transformation. This construction results in a new
representation of the W-infinity algebras in terms of 4 bosonic fields.Comment: 13 pages, Latex, CERN-TH.6627/9
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