6 research outputs found
Yang-Baxter maps and multi-field integrable lattice equations
A variety of Yang-Baxter maps are obtained from integrable multi-field
equations on quad-graphs. A systematic framework for investigating this
connection relies on the symmetry groups of the equations. The method is
applied to lattice equations introduced by Adler and Yamilov and which are
related to the nonlinear superposition formulae for the B\"acklund
transformations of the nonlinear Schr\"odinger system and specific
ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio
Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices
A family of nonparametric Yang Baxter (YB) maps is constructed by
refactorization of the product of two 2 by 2 matrix polynomials of first
degree. These maps are Poisson with respect to the Sklyanin bracket. For each
Casimir function a parametric Poisson YB map is generated by reduction on the
corresponding level set. By considering a complete set of Casimir functions
symplectic multiparametric YB maps are derived. These maps are quadrirational
with explicit formulae in terms of matrix operations. Their Lax matrices are,
by construction, 2 by 2 first degree polynomial in the spectral parameter and
are classified by Jordan normal form of the leading term. Nonquadrirational
parametric YB maps constructed as limits of the quadrirational ones are
connected to known integrable systems on quad graphs
Quantum discrete Dubrovin equations
The discrete equations of motion for the quantum mappings of KdV type are
given in terms of the Sklyanin variables (which are also known as quantum
separated variables). Both temporal (discrete-time) evolutions and spatial
(along the lattice at a constant time-level) evolutions are considered. In the
classical limit, the temporal equations reduce to the (classical) discrete
Dubrovin equations as given in a previous publication. The reconstruction of
the original dynamical variables in terms of the Sklyanin variables is also
achieved.Comment: 25 page
T-systems and Y-systems in integrable systems
The T and Y-systems are ubiquitous structures in classical and quantum
integrable systems. They are difference equations having a variety of aspects
related to commuting transfer matrices in solvable lattice models, q-characters
of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras
with coefficients, periodicity conjectures of Zamolodchikov and others,
dilogarithm identities in conformal field theory, difference analogue of
L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem,
AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace
sequence in discrete geometry, Fermionic character formulas and combinatorial
completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics,
analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and
so forth. This review article is a collection of short reviews on these topics
which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5,
eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical
review) also needs these correction