Related papers: Integrable systems on hexagonal lattices and consi…

Hex systems were recently introduced [A. P. Kels. Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces. 2022. arXiv:2205.02720 [math-ph]] as systems of equations defined on…

There is a recently discovered formulation of the multidimensional consistency integrability condition for lattice equations, called consistency-around-a-face-centered-cube(CAFCC), which is applicable to equations defined on a vertex and…

A new set of discrete integrable equations, called face-centered quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang-Baxter equation. These equations satisfy a new formulation of…

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1.…

Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…

There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations which satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former.…

For two-dimensional lattice equations one definition of integrability is that the model can be naturally and consistently extended to three dimensions, i.e., that it is "consistent around a cube" (CAC). As a consequence of CAC one can…

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…

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional…

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…

The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The…

We consider overdetermined systems of difference equations for a single function $u$ which are consistent, and propose a general framework for their analysis. The integrability of such systems is defined as the existence of higher order…

Multidimensional Consistency becomes more and more important in the theory of discrete integrable systems. Recently, we gave a classification of all 3D consistent 6-tuples of equations with the tetrahedron property, where several novel…

We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability…

The paper proposes a novel hybrid method for solving equilibrium problems and fixed point problems. By constructing specially cutting-halfspaces, in this algorithm, only an optimization program is solved at each iteration without the…

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and…

In this paper we define the algebraic entropy test for face-centered quad equations, which are equations defined on vertices of a quadrilateral plus an additional interior vertex. This notion of algebraic entropy is applied to a recently…

A cascade of phase transitions from square to hexagonal lattice is studied in 2D system of particles interacting via core-softened potential. Due to the presence of two length-scales of repulsion, different local configurations with four,…

For the 5-components Maxwell-Bloch system the stability problem for the isolated equilibria is completely solved. Using the geometry of the symplectic leaves, a detailed construction of the homoclinic orbits is given. Studying the problem…

There exist cubical transition systems containing cubes having an arbitrarily large number of faces. A regular transition system is a cubical transition system such that each cube has the good number of faces. The categorical and…