Yang-Baxter maps and symmetries of integrable equations on quad-graphs

Abstract

A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multi-parameter symmetry groups of the equations. We use the classification results by Adler, Bobenko and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multi-field integrable equations.

20 pages, 5 figures

References in corpus (2)

• Yang-Baxter maps: dynamical point of view
• The Boussinesq integrable system. Compatible lattice and continuum structures

Cited by in corpus (59)

• Yang-Baxter maps: dynamical point of view
• On Quadrirational Yang-Baxter Maps
• Darboux transformations, finite reduction groups and related Yang-Baxter maps
• On non-multiaffine consistent-around-the-cube lattice equations
• Integrable Matrix Models in Discrete Space-Time
• Non-commutative rational Yang-Baxter maps
• Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices
• Yang-Baxter maps and multi-field integrable lattice equations
• Set-theoretical reflection equation: Classification of reflection maps
• Affine linear and D4 symmetric lattice equations : symmetry analysis and reductions
• A noncommutative discrete potential KdV lift
• On non-abelian quadrirational Yang-Baxter maps
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Yang-Baxter maps associated to elliptic curves
• Families of Integrable Equations
• Poisson Yang-Baxter maps with binomial Lax matrices
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Some properties of the k-dimensional Lyness' map
• The pentagon relation and incidence geometry
• On the $3D$ consistency of a Grassmann extended lattice Boussinesq system
• On discrete integrable equations of higher order
• Entwining Yang-Baxter maps related to NLS type equations
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups
• Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method
• Relativistic Collisions as Yang-Baxter maps
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• Commutation Relations and Discrete Garnier Systems
• Classification of 3-dimensional integrable scalar discrete equations
• The Coxeter relations and KP map for non-commuting symbols
• Discrete Integrable Equations over Finite Fields
• 3D compatible ternary systems and Yang-Baxter maps
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Integrable Boundary for Quad-Graph Systems: Three-Dimensional Boundary Consistency
• Quadrirational Yang-Baxter maps and the affine-E8 Painleve lattice
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Yang-Baxter Maps from the Discrete BKP Equation
• Two-component Yang-Baxter maps and star-triangle relations
• Systems of difference equations on a vector valued function that admit 3D space of scalar potentials
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• Multi-quadratic quad equations: integrable cases from a factorised-discriminant hypothesis
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings
• Integrable multi-component difference systems of equations
• Reflection groups and discrete integrable systems
• On the singularity structure of a discrete modified-Korteweg-deVries equation
• Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps
• Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
• Soliton solutions of a generalized discrete KdV equation
• Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map
• ${\mathbb{Z}}_N$ graded discrete Lax pairs and Yang-Baxter maps
• Self-distributive structures, braces & the Yang-Baxter equation
• Classification of integrable boundary equations for integrable quad-graph systems
• Integrable extensions of Adler's map via Grassmann algebras
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps
• $2^n-$rational maps