Related papers: Integrable systems on quad-graphs

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 extend integrable systems on quad-graphs, such as the Hirota equation and the cross-ratio equation, to the non-commutative context, when the fields take values in an arbitrary associative algebra. We demonstrate that the…

Derivation-based differential calculi are of great importance in noncommutative geometry, noncommutative gauge theory and integrable systems. In this paper, we propose the connection and curvature from a class of deformed derivation-based…

For all non-symmetric discrete relativistic Toda type equations we establish a relation to 3D consistent systems of quad-equations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of $\mathbb…

We propose the notion of integrable boundary in the context of discrete integrable systems on quad-graphs. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called…

In the context of integrable partial difference equations on quad-graphs, we introduce the notion of open boundary reductions as a new means to construct discrete integrable mappings and their invariants. This represents an alternative to…

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…

One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitute the models with the spectral parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov…

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and…

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…

For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a function $A(\lambda)$. Those integrable systems provide an…

We review recent results on Integrable Discrete Geometry. It turns out that most of the known (continuous and/or discrete) integrable systems are particular symmetries of the quadrilateral lattice, a multidimensional lattice characterized…

We present a slight generalization of the notion of completely integrable systems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.

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…

For networks of coupled dynamical systems we characterize admissible functions, that is, functions whose gradient is an admissible vector field. The schematic representation of a gradient network dynamical system is of an undirected cell…

In this paper we introduce the notion of coalgebra symmetry for discrete systems. With this concept we prove that all discrete radially symmetric systems in standard form are quasi-integrable and that all variational discrete quasi-radially…

Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in…

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric…

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their…

The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and…