Related papers: Lattice equations arising from discrete Painlev\'e…
A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional…
Building on the earlier work of Odeh, Barany, Golubitsky, Turski and Lasher we give a proof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model of superconductivity.
We consider 3D consistent systems of six independent quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of…
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on…
We present the discrete, q-, form of the Painlev\'e VI equation written as a three-point mapping and analyse the structure of its singularities. This discrete equation goes over to P_{VI} at the continuous limit and degenerates towards the…
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of…
Lattice sums of cuboidal lattices, which connect the face-centered with the mean-centered and the body-centered cubic lattices through parameter dependent lattice vectors, are evaluated by decomposing them into two separate lattice sums…
We give a simple form of the evolution equation and a scalar Lax pair for the $q$-Garnier system. Some reductions to the $q$-Painlev\'e equations and the autonomous case as a generalized QRT system are discussed. Using two kinds of Pad\'e…
We consider the symmetric q-Painlev\'e equations derived from the birational representation of affine Weyl groups by applying the projective reduction and construct the hypergeometric solutions. Moreover, we discuss continuous limits of the…
We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-\beta. We combine both lattices to obtain the bi-lattice N \cup (N+1-\beta) and show that the orthogonal polynomials…
In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlev\'e equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that…
In this paper, we derive the explicit formula for higher cup products on hypercubic lattices, based on the recently developed geometrical interpretation in the simplicial case. We illustrate how this formalism can elucidate lattice…
In this article an other equivalent linear representation of classical Painlev\'e second equation is derived by introducing a gauge transformation to old Lax pair. The new linear system of that equation carries similar structure as other…
We present an extension of the framework introduced in [1] to treat multicomponent systems, showing that new degrees of freedom are necessary in order to obtain the desired boundary conditions. We then apply this extended framework to the…
In the paper [V. Adler, IMRN {\bf 1} (1998) 1--4] a lattice version of the Krichever-Novikov equation was constructed. We present in this note its Lax pair and discuss its elliptic form.
Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the…
In this paper, we present multi parametric quadgraph equations which are consistent around the cube. These equations are obtained by applying a `double twist' to known integrable equations. Furthermore, we perform a limit to one of these…
We point out two extensions of the relation between matrix models, topological strings and N=1 supersymmetric gauge theories. First, we note that by considering double scaling limits of unitary matrix models one can obtain large N duals of…
The low-energy properties of the SU(4) spin-orbital model on a two-leg ladder are studied by a variety of analytical and numerical techniques. Like in the case of SU(2) models, there is a singlet-multiplet gap in the spectrum, but the…
We present a discrete inverse scattering transform for all ABS equations excluding Q4. The nonlinear partial difference equations presented in the ABS hierarchy represent a comprehensive class of scalar affine-linear lattice equations which…