543 research outputs found
Commuting difference operators with elliptic coefficients from Baxter's vacuum vestors
For quantum integrable models with elliptic R-matrix, we construct the Baxter
Q-operator in infinite-dimensional representations of the algebra of
observables.Comment: 31 pages, LaTeX, references adde
SDiff(2) Toda equation -- hierarchy, function, and symmetries
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda
equation, is shown to have a Lax formalism and an infinite hierarchy of higher
flows. The Lax formalism is very similar to the case of the self-dual vacuum
Einstein equation and its hyper-K\"ahler version, however now based upon a
symplectic structure and the group SDiff(2) of area preserving diffeomorphisms
on a cylinder . An analogue of the Toda lattice tau function is
introduced. The existence of hidden SDiff(2) symmetries are derived from a
Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function
turn out to have commutator anomalies, hence give a representation of a central
extension of the SDiff(2) algebra.Comment: 16 pages (``vanilla.sty" is attatched to the end of this file after
``\bye" command
Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
We discuss an algebraic method for constructing eigenvectors of the transfer
matrix of the eight vertex model at the discrete coupling parameters. We
consider the algebraic Bethe ansatz of the elliptic quantum group for the case where the parameter satisfies for arbitrary integers , and . When or
is odd, the eigenvectors thus obtained have not been discussed previously.
Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin
chain, some of which are shown to be related to the loop algebra
symmetry of the XXZ spin chain. We show that the dimension of some degenerate
eigenspace of the XYZ spin chain on sites is given by , if
is an even integer. The construction of eigenvectors of the transfer matrices
of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices
hbar-Dependent KP hierarchy
This is a summary of a recursive construction of solutions of the
hbar-dependent KP hierarchy. We give recursion relations for the coefficients
X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +
... for which the dressing operator W is expressed in the exponential form W =
\exp(X/\hbar). The asymptotic behaviours of (the logarithm of) the wave
function and the tau function are also considered.Comment: 12 pages, contribution to the Proceedings of the "International
Workshop on Classical and Quantum Integrable Systems 2011" (January 24-27,
2011 Protvino, Russia
Remarks on the waterbag model of dispersionless Toda Hierarchy
We construct the free energy associated with the waterbag model of dToda.
Also, the relations of conserved densities are investigatedComment: 12 page
Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
This paper reconsiders finite variable reductions of the universal Whitham
hierarchy of genus zero in the perspective of dispersionless Hirota equations.
In the case of one-variable reduction, dispersionless Hirota equations turn out
to be a powerful tool for understanding the mechanism of reduction. All
relevant equations describing the reduction (L\"owner-type equations and
diagonal hydrodynamic equations) can be thereby derived and justified in a
unified manner. The case of multi-variable reductions is not so
straightforward. Nevertheless, the reduction procedure can be formulated in a
general form, and justified with the aid of dispersionless Hirota equations. As
an application, previous results of Guil, Ma\~{n}as and Mart\'{\i}nez Alonso
are reconfirmed in this formulation.Comment: latex 2e using packages amsmath,amssymb,amsthm, 39 pages, no figure;
(v2) a few typos corrected and accepted for publicatio
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
Explorations of the Extended ncKP Hierarchy
A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy
(ncKP hierarchy) by a set of evolution equations in the Moyal-deformation
parameters is further explored. Formulae are derived to compute these equations
efficiently. Reductions of the xncKP hierarchy are treated, in particular to
the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of
the Sato formalism for the KP hierarchy is carried over to the generalized
framework. In particular, the well-known bilinear identity theorem for the KP
hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends
to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions
of the ncKP equation are also solutions of the first few deformation equations.
This is shown to be related to the existence of certain families of algebraic
identities.Comment: 34 pages, correction of typos in (7.2) and (7.5
Integrable (2+1)-dimensional systems of hydrodynamic type
We describe the results that have so far been obtained in the classification
problem for integrable (2+1)-dimensional systems of hydrodynamic type. The
systems of Gibbons--Tsarev type are the most fundamental here. A whole class of
integrable (2+1)-dimensional models is related to each such system. We present
the known GT systems related to algebraic curves of genus g=0 and g=1 and also
a new GT system corresponding to algebraic curves of genus g=2. We construct a
wide class of integrable models generated by the simplest GT system, which was
not considered previously because it is in a sense trivial.Comment: 47 pages, no figure
The master T-operator for vertex models with trigonometric -matrices as classical tau-function
The construction of the master T-operator recently suggested in Alexandrov et
al. (arXiv:1112.3310) is applied to integrable vertex models and associated
quantum spin chains with trigonometric R-matrices. The master T-operator is a
generating function for commuting transfer matrices of integrable vertex models
depending on infinitely many parameters. At the same time it turns out to be
the tau-function of an integrable hierarchy of classical soliton equations in
the sense that it satisfies the the same bilinear Hirota equations. The class
of solutions of the Hirota equations that correspond to eigenvalues of the
master T-operator is characterized and its relation to the classical
Ruijsenaars-Schneider system of particles is discussed.Comment: 19 pages, for proceedings of the workshop "Classical and Quantum
Integrable Systems" (Dubna, 23-27 January 2012), typos correcte
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