1,303 research outputs found
Poisson Yang-Baxter maps with binomial Lax matrices
A construction of multidimensional parametric Yang-Baxter maps is presented.
The corresponding Lax matrices are the symplectic leaves of first degree matrix
polynomials equipped with the Sklyanin bracket. These maps are symplectic with
respect to the reduced symplectic structure on these leaves and provide
examples of integrable mappings. An interesting family of quadrirational
symplectic YB maps on with Lax
matrices is also presented.Comment: 22 pages, 3 figure
Classical R-Operators and Integrable Generalizations of Thirring Equations
We construct different integrable generalizations of the massive Thirring
equations corresponding loop algebras in
different gradings and associated ''triangular'' -operators. We consider the
most interesting cases connected with the Coxeter automorphisms, second order
automorphisms and with ''Kostant-Adler-Symes'' -operators. We recover a
known matrix generalization of the complex Thirring equations as a partial case
of our construction.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Solution of the classical Yang--Baxter equation with an exotic symmetry, and integrability of a multi-species boson tunneling model
Solutions of the classical Yang-Baxter equation provide a systematic method
to construct integrable quantum systems in an algebraic manner. A Lie algebra
can be associated with any solution of the classical Yang--Baxter equation,
from which commuting transfer matrices may be constructed. This procedure is
reviewed, specifically for solutions without skew-symmetry. A particular
solution with an exotic symmetry is identified, which is not obtained as a
limiting expansion of the usual Yang--Baxter equation. This solution
facilitates the construction of commuting transfer matrices which will be used
to establish the integrability of a multi-species boson tunneling model. The
model generalises the well-known two-site Bose-Hubbard model, to which it
reduces in the one-species limit. Due to the lack of an apparent reference
state, application of the algebraic Bethe Ansatz to solve the model is
prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of
operator identities and tensor product decompositions.Comment: 22 pages, no figure
New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum
quasi-Yang-Baxter algebras being simple but non-trivial deformations of
ordinary algebras of monodromy matrices realize a new type of quantum dynamical
symmetries and find an unexpected and remarkable applications in quantum
inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter
algebras the standard procedure of QISM one obtains new wide classes of quantum
models which, being integrable (i.e. having enough number of commuting
integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic
Bethe ansatz solution for arbitrarily large but limited parts of the spectrum).
These quasi-exactly solvable models naturally arise as deformations of known
exactly solvable ones. A general theory of such deformations is proposed. The
correspondence ``Yangian --- quasi-Yangian'' and `` spin models ---
quasi- spin models'' is discussed in detail. We also construct the
classical conterparts of quasi-Yang-Baxter algebras and show that they
naturally lead to new classes of classical integrable models. We conjecture
that these models are quasi-exactly solvable in the sense of classical inverse
scattering method, i.e. admit only partial construction of action-angle
variables.Comment: 49 pages, LaTe
Classical and Quantum Nonultralocal Systems on the Lattice
We classify nonultralocal Poisson brackets for 1-dimensional lattice systems
and describe the corresponding regularizations of the Poisson bracket relations
for the monodromy matrix . A nonultralocal quantum algebras on the lattices for
these systems are constructed.For some class of such algebras an
ultralocalization procedure is proposed.The technique of the modified
Bethe-Anzatz for these algebras is developed.This technique is applied to the
nonlinear sigma model problem.Comment: 33 pp. Latex. The file is resubmitted since it was spoiled during
transmissio
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