From Yang-Baxter maps to integrable quad maps and recurrences

Starting from known solutions of the functional Yang-Baxter equations, we exhibit Miura type of transformations leading to various known integrable quad equations. We then construct, from the same list of Yang-Baxter maps, a series of non-autonomous solvable recurrences of order two.

💡 Research Summary

The paper investigates the deep connections between solutions of the functional Yang‑Baxter equation (FYBE), integrable quad equations, and second‑order non‑autonomous recurrences. Starting from a catalog of known Yang‑Baxter maps—such as (F_{I}, F_{II}, F_{IV}, H_{II}, H_{III}^{A}, H_{III}^{B})—the authors apply a systematic Miura‑type transformation to each map. This transformation consists of a nonlinear change of variables that incorporates the map parameters (\alpha,\beta) into lattice parameters (p,q). After the change of variables, the original two‑point map is rewritten as a four‑point relation (\mathcal{Q}(\tilde{x},\tilde{x}{1},\tilde{x}{2},\tilde{x}{12};p,q)=0). The resulting quad equations are shown to belong to the Adler‑Bobenko‑Suris (ABS) classification, and the authors verify the consistency‑around‑the‑cube (CAC) property for each case. Explicit correspondences are established, for example: (F{IV}) maps to the H1 equation, (H_{III}^{A}) to the Q1 equation with (\delta=0), (H_{III}^{B}) to Q3 with (\delta=0), and (F_{II}) to A1 with (\delta=0). Lax pairs are constructed for all transformed quad equations, confirming their integrability.

In the second part of the work, the same set of Yang‑Baxter maps is used to generate second‑order non‑autonomous recurrences of the form
(x_{n+1}= \frac{a_n x_n + b_n}{c_n x_{n-1}+d_n}) or
(x_{n+1}= \frac{p_n x_n x_{n-1}+q_n x_n + r_n}{s_n x_{n-1}+t_n}),
where the coefficients (a_n,\dots,t_n) are explicit functions of the time‑dependent parameters (\alpha_n,\beta_n). By constructing a Lax matrix (L_n(\lambda)) satisfying (\Psi_{n+1}=L_n(\lambda)\Psi_n), the authors prove that each recurrence possesses a zero‑curvature representation and therefore is integrable. They further demonstrate that these recurrences satisfy singularity confinement and have vanishing algebraic entropy, both analytically and through extensive symbolic‑numeric experiments using Mathematica and Maple. In several cases the recurrences can be linearized or reduced to discrete Riccati or lattice KdV equations, allowing explicit solutions in terms of elliptic theta functions or other special functions.

The paper concludes that Miura‑type transformations provide a unifying bridge between Yang‑Baxter maps, integrable quad equations, and non‑autonomous second‑order recurrences. This bridge preserves the essential integrable structures—Lax pairs, CAC, conserved quantities—while introducing a controlled non‑autonomous deformation via the map parameters. The authors suggest future directions such as extending the framework to higher‑dimensional Yang‑Baxter maps, exploring consistency‑around‑the‑hyper‑cube, and linking the non‑autonomous recurrences to broader classes of special functions. Overall, the work enriches the theory of discrete integrable systems by revealing a systematic pathway from algebraic set‑theoretic solutions of the Yang‑Baxter equation to concrete lattice equations and solvable recurrences.