Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps

Abstract

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the $H_{\mathrm{I}}$, $H_{\mathrm{II}}$ and $H_{\mathrm{III}}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_{\mathrm{I}}$, $H_{\mathrm{II}}$ and $H_{\mathrm{III}}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H$-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H$-list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to $k$-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.

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• On non-abelian quadrirational Yang-Baxter maps
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Entwining Yang-Baxter maps related to NLS type equations
• 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
• Matrix factorizations and pentagon maps
• Soliton interactions and Yang-Baxter maps for the complex coupled short-pulse equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Two-component Yang-Baxter maps and star-triangle relations
• Discrete Lax pairs and hierarchies of integrable difference systems
• Integrable multi-component difference systems of equations