Related papers: On multidimensional consistent systems of asymmetr…
Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to…
We prove the Bianchi permutability (existence of superposition principle) of B\"acklund transformations for asymmetric quad-equations. Such equations and there B\"acklund transformations form 3D consistent systems of a priori different…
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…
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…
A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter…
Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and…
We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the…
The new concept of a system of hex equations is introduced as an overdetermined system of six five-point face-centered quad equations defined on six vertices of a hexagon. For a consistent system of hex equations, two variables on…
The main goal of this paper is to find the discrete analogue of the Bianchi system in spaces of arbitrary dimesion together with its geometric interpretation. We show that the proper geometric framework of such generalization is the…
We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion…
Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in $\mathbb Z^d$. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of…
In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the…
We classify multidimensionally consistent maps given by (formal or convergent) series of the following kind: $$ T_k x_{ij}=x_{ij} + \sum_{m=2}^\infty A_{ij ; \, k}^{(m)}(x_{ij},x_{ik},x_{jk}), $$ where $A_{ij;\, k}^{(m)}$ are homogeneous…
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…
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…
We study 2D discrete integrable equations of order 1 with respect to one independent variable and $m$ with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The…
We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail, and…
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 has emerged as a key integrability property for partial difference equations (P$\Delta$Es) defined on the "space-time" lattice. It has led, among other major insights, to a classification of scalar affine-linear…
In this paper an approach to generate multi-dimensionally consistent $N$-component systems is proposed. The approach starts from scalar multi-dimensionally consistent quadrilateral systems and makes use of the cyclic group. The obtained…