Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps

Abstract

We present two non-equivalent families of hierarchies of non-Abelian compatible maps and we provide their Lax pair formulation. These maps are associated with families of hierarchies of non-Abelian Yang-Baxter maps, which we provide explicitly. In addition, these hierarchies correspond to integrable difference systems with variables defined on edges of an elementary cell of the ${\mathbb{Z}}^2$ graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. In that respect we obtain the non-Abelian lattice-modified Gel'fand-Dikii hierarchy, together with the explicit form of a non-Abelian hierarchy that we refer to as the lattice-NQC (or lattice-$(Q3{)}_0$) Gel'fand-Dikii hierarchy.

arXiv comment29 pages, 2 figures. v2:Typos corrected, one figure added

References in corpus (36)

• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever-Novikov) System
• Yang-Baxter maps and integrable dynamics
• Yang-Baxter maps and symmetries of integrable equations on quad-graphs
• Classification of 3D consistent quad-equations
• Lax matrices for Yang-Baxter maps
• Boussinesq-like multi-component lattice equations and multi-dimensional consistency
• On Quadrirational Yang-Baxter Maps
• The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice
• On non-multiaffine consistent-around-the-cube lattice equations
• Grassmann extensions of Yang-Baxter maps
• Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
• Non-commutative rational Yang-Baxter maps
• Symmetries and integrability of discrete equations defined on a black-white lattice
• A noncommutative discrete potential KdV lift
• On non-abelian quadrirational Yang-Baxter maps
• An integrable multicomponent quad equation and its Lagrangian formulation
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Non-commutative lattice modified Gel'fand-Dikii systems
• Cauchy problem for integrable discrete equations on quad-graphs
• Multi-Component Extension of CAC Systems
• Families of Integrable Equations
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• On the integrability of a new lattice equation found by multiple scale analysis
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• The Coxeter relations and KP map for non-commuting symbols
• The discrete potential Boussinesq equation and its multisoliton solutions
• Non-commutative Hermite--Padé approximation and integrability
• A higher-rank version of the Q3 equation
• Grassmannian-parameterized solutions to direct-sum polygon and simplex equations
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• Yang-Baxter Maps from the Discrete BKP Equation
• Discrete Lax pairs and hierarchies of integrable difference systems
• Non-commutative q-Painleve VI equation
• Lattice modified KdV hierarchy from a Lax pair expansion
• On generalized Lotka-Volterra lattices

Cited by in corpus (2)

• Matrix factorizations and pentagon maps
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps