Integrable systems on quad-graphs

By Alexander I. Bobenko, Yuri B. Suris. 2001

317 citations recorded by the source corpus.

Discrete (lattice) systems constitute a well-established part of the theory of integrable systems. They came up already in the early days of the theory (see, e.g. [11, 12]), and took gradually more and more important place in it (cf. a review in [18]). Nowadays many experts in the field agree that discrete integrable systems are in many respects even more fundamental than the continuous ones. They play a prominent role in various applications of integrable systems such as discrete differential geometry (see, e.g., a review in [9]). Traditionally, independent variables of discrete integrable systems are considered as belonging to a regular square lattice Z (or its multidimensional analogs Z). Only very recently, there appeared first hints on the existence of a rich and meaningful theory of integrable systems on nonsquare lattices and, more generally, on arbitrary graphs. The relevant publications are almost exhausted by [2, 3, 5, 6, 16, 20, 21, 22]. We define integrable systems on graphs as flat connections with the values in loop groups. This is very natural definition, and experts in discrete integrable systems will not only immediately accept it, but might even consider it trivial. Nevertheless, it crystallized only very recently, and seems not to appear in the literature before [3, 5, 6]. (It should be noted that a different framework for integrable systems on graphs is being developed by Novikov with collaborators [16, 20, 21].) We were led to considering such systems by our (with Hoffmann) investigations of circle patterns as objects of discrete complex analysis: in [5, 6] we demonstrated that certain classes of circle patterns with the combinatorics of regular hexagonal lattice

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/50df85408bee01b43bd8da0d53cb7058765eccff

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Citations in the corpus

Soliton solutions for ABS lattice equations: I. Cauchy matrix approach (2009, background): “[24, 11], where p and q play the role of lattice parameters.”
Geometry of the Yang-Baxter Maps: pencils of conics and quadrirational mappings (2003, unknown): “[4] A.”
Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed Heisenberg Algebras (2003, methodology): “it, which emphasizes the relation with 3D consistency condition for discrete equations on quad-graphs (see [3, 5]).”
Symmetries of Integrable Difference Equations on the Quad‐Graph (2007, unknown): “Since then, Lax pairs have been derived for these and many other quad-graph equations [6, 7, 21].”
Discrete nonlinear hyperbolic equations. Classification of integrable cases (2007, background): “For d = 2, a decisive step was made in [10] and independently in [19]: it was shown that the integrability in the usual sense of soliton theory (as the existence of a zero curvature representation) follows for two-dimensional systems from the three-dimensional consistency.”
Lax pair for the Adler (lattice Krichever–Novikov) system (2001, background): “directions, cf. [21] for details, (in the paper [14] we gave the lattice modiﬁed KdV equation as an example, where the scheme works precisely in the same way). We mention that in a recent submission, [22], the consistency around the cube was reiterated in a slightly diﬀerent context. The relevant fo”
Variational principles for circle patterns and Koebe’s theorem (2002, background): “They correspond to the edges of �. There are two kinds of vertices of �q, ‘white’ vertices corresponding to faces of � and ‘black’ vertices corresponding to vertices of �. Quad-graphs turn out to be crucial for defining discrete integrable systems [ BS02 ] (in particular those related to circle patt”
Classification of integrable equations on quad-graphs. The consistency approach (2002, methodology): “A Lax representation for (Q4) was found in [26] with the help of the method based on the three–dimensional consistency, identical with the method introduced in [8].”