Yang-Baxter Maps from the Discrete BKP Equation

We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

💡 Research Summary

The paper investigates the relationship between the discrete BKP (Bogoyavlensky‑Kadomtsev‑Petviashvili) equation and set‑theoretical solutions of the Yang‑Baxter equation (YBE). By imposing an N‑reduction (τ_{k+N}=τ_k) on the Hirota‑bilinear form of the discrete BKP equation, the authors obtain a finite‑dimensional dynamical system whose variables are expressed as ratios of τ‑functions. This reduction yields explicit rational transformation formulas that map a pair of variables (x, y) to a new pair (x′, y′). The resulting map, denoted R_N, is shown to satisfy the set‑theoretical YBE, R_{12} ∘ R_{13} ∘ R_{23}=R_{23} ∘ R_{13} ∘ R_{12}, for any positive integer N. The proof relies on the Plücker relations satisfied by the τ‑functions and on a careful algebraic manipulation that demonstrates the associativity of the three‑body interaction encoded by R_N. Importantly, the construction is uniform: the same algebraic steps work for arbitrary N, thereby providing a whole family of rational Yang‑Baxter maps that generalize previously known quadrirational maps (the case N = 2).

Having established the rational maps, the authors turn to their tropical (ultra‑discrete) limit. By introducing logarithmic variables and letting a scaling parameter ε tend to zero, the rational expressions are replaced by piecewise‑linear formulas involving the max‑plus algebra. The limiting map, called T_N, retains the YBE property; its verification follows from the tropical analogue of the Plücker relations. T_N is thus a new class of piecewise‑linear Yang‑Baxter maps, distinct from earlier tropical maps because it inherits the N‑reduction structure of the BKP hierarchy.

The paper also addresses several structural aspects of both families of maps. First, invertibility is proved: explicit inverse formulas are derived by solving the τ‑function relations backward, guaranteeing that both R_N and T_N are bijections on the appropriate domains. Second, invariants are identified. For the rational maps, a multiplicative invariant of the form ∏_{i=1}^N f(x_i) (with f a rational function) is conserved under iteration, while for the tropical maps a “tropical determinant”—a max‑plus expression built from the variables—is invariant. These conserved quantities signal the integrability of the maps and connect them to the underlying discrete BKP dynamics.

In the final section, the authors discuss connections to known integrable lattice models. When N = 2, R_N reduces to the well‑studied quadrirational Yang‑Baxter map, and T_N reduces to its tropical counterpart. For N > 2 the maps describe genuinely higher‑dimensional interactions, suggesting potential applications to multi‑component lattice systems, higher‑rank quantum groups, and combinatorial models such as box‑ball systems. Moreover, the tropical maps provide a bridge to tropical geometry, offering a geometric interpretation of the piecewise‑linear dynamics.

Overall, the work delivers a systematic method to generate both rational and piecewise‑linear Yang‑Baxter maps from the discrete BKP equation for any reduction size N. It enriches the catalogue of set‑theoretical YBE solutions, demonstrates their integrable structure through invariants and invertibility, and opens avenues for applying these maps in statistical mechanics, quantum integrable systems, and tropical combinatorics.