1,153 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
Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems
A systematic construction of St\"{a}ckel systems in separated coordinates and
its relation to bi-Hamiltonian formalism are considered. A general form of
related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is
derived. One Casimir bi-Hamiltonian case is studed in details and in this case,
a systematic construction of related hydrodynamic systems in arbitrary
coordinates is presented, using a cofactor method and soliton symmetry
constraints.Comment: to appear in Journal of Geometry and Physic
Functional representation of the Ablowitz-Ladik hierarchy
The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the
inverse scattering approach. After establishing the structure of solutions of
the auxiliary linear problems, the ALH, which has been originally introduced as
an infinite system of difference-differential equations is presented as a
finite system of difference-functional equations. The representation obtained,
when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate
relations between the ALH and some other integrable systems, the
Kadomtsev-Petviashvili hierarchy in particular.Comment: 15 pages, LaTe
On deformations of standard R-matrices for integrable infinite-dimensional systems
Simple deformations, with a parameter , of classical -matrices
which follow from decomposition of appropriate Lie algebras, are considered. As
a result nonstandard Lax representations for some well known integrable systems
are presented as well as new integrable evolution equations are constructed.Comment: 14 page
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