The set-theoretical reflection equation is the classical counterpart of the quantum reflection equation governing integrable boundaries; it was introduced by Caudrelier and Zhang from the factorization of vector-soliton interactions on the half-line, and its solutions (reflection maps) were classified for all quadrirational Yang-Baxter map families by Caudrelier, Crampe and Zhang (2013), whose conclusions posed six directions for future work. Three of these directions have since been substantially settled: the meaning of reflection maps for 3D consistency on quad-graphs (3D boundary consistency and the classification of integrable boundary equations; refs [11], [19], [21], [23]), systematic algebraic constructions of reflection maps via braces, racks and crystals (refs [13], [14], [16]-[18], [20], [24], [26], [29]), and dynamical reflection maps (ref [28]). The unresolved open core comprises three linked components: (A) a geometric reinterpretation of the quadrirational reflection-map classification in the spirit of the pencil-of-conics geometry of Adler-Bobenko-Suris; (B) determining, for fully discrete integrable systems with integrable boundary - ABS quad-graph equations on a half-plane or the discrete vector Ablowitz-Ladik (Manakov) system on a half-lattice - whether soliton-boundary interactions are encoded by a reflection map on the discrete soliton data; and (C) the set-theoretical analog of the Freidel-Maillet quadratic boundary algebras involving four objects.

Background

The Yang-Baxter equation is the master algebraic relation of integrable models; in its set-theoretical form its solutions are bijections R:S×S→S×SR:S\times S\to S\times S (with SS a set), called Yang-Baxter maps, satisfying R12R13R23=R23R13R12R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12} as an identity on S×S×SS\times S\times S, where RijR_{ij} acts as RR on the ii-th and jj-th copies. Yang-Baxter maps govern the refactorization of soliton dressing transformations, and they are one facet of discrete integrability: on a quad-graph (a lattice of quadrilaterals on which partial difference equations live), integrability is characterized by three-dimensional (3D) consistency of the equation around a cube. A map on CP1×CP1\mathbb{CP}^1\times\mathbb{CP}^1 is called quadrirational when the two components of the map and the two components of its inverse are all rational; Adler, Bobenko and Suris classified the quadrirational Yang-Baxter maps into the families FIF_I-FVF_V, together with degenerate maps, and gave the classification a geometric meaning: each map is encoded by a pencil of conics (two conics intersecting in the four points that determine the correspondence), so the classification becomes a statement about pencils of conics. Papageorgiou, Suris, Tongas and Veselov subsequently completed the quadrirational classification, which also comprises the symmetry-related families HIH_I-HVH_V. In parallel, quantum integrable systems with boundaries are governed by a second relation, the reflection (or boundary) equation, introduced by Cherednik and developed by Sklyanin; its solutions are K-matrices encoding how particles reflect off an integrable boundary. Caudrelier and Zhang, studying the vector nonlinear Schrödinger equation (the Manakov system) on the half-line with an integrable boundary via the nonlinear mirror image method (which extends an integrable solution to the whole line by an involution and reads off the boundary interaction), discovered that the factorization of soliton-soliton and soliton-boundary interactions is encoded by a map on the soliton polarization data, which they called a reflection map: given a Yang-Baxter map RR, a reflection map BB satisfies the set-theoretical reflection equation

B1R21B2R12=R21B2R12B1,B_1R_{21}B_2R_{12}=R_{21}B_2R_{12}B_1,

where BiB_i acts as BB on the ii-th copy of SS. For parametric Yang-Baxter maps R12(a,b)R_{12}(a,b) depending on parameters a,ba,b from a set Σ\Sigma (as soliton interactions do), a reflection map acts as B:(X,a)↦(ha(X),σ(a))B:(X,a)\mapsto(h_a(X),\sigma(a)) with a parameter map σ\sigma and satisfies the parametric version

B1(a) R21(σ(b),a) B2(b) R12(a,b)=R21(σ(b),σ(a)) B2(b) R12(σ(a),b) B1(a).B_1(a)\,R_{21}(\sigma(b),a)\,B_2(b)\,R_{12}(a,b)=R_{21}(\sigma(b),\sigma(a))\,B_2(b)\,R_{12}(\sigma(a),b)\,B_1(a).

Caudrelier, Crampe and Zhang developed the general theory and, by a folding technique (identifying the two copies of SS through an involution φa\varphi_a and imposing b=σ(a)b=\sigma(a)), classified the reflection maps associated to all quadrirational Yang-Baxter map families. Their conclusions posed six directions: (i) a geometric reinterpretation of this classification along the lines of the pencil-of-conics arguments; (ii) general, abstract constructions of reflection maps from the known general constructions of Yang-Baxter maps; (iii) the meaning of reflection maps for 3D consistency on quad-graphs; (iv) whether the soliton-boundary interpretation extends to fully discrete integrable systems; (v) set-theoretical analogs of the quantum generalizations of the reflection equation, namely the quadratic algebras of Freidel and Maillet (in which four matrices obey Yang-Baxter-type relations, generalizing the single-matrix reflection equation) and reflection-type equations associated to dynamical Yang-Baxter maps (set-theoretical maps, introduced by Shibukawa, that depend on a parameter which the map itself transforms, subject to an invariance condition); and (vi) applications to quantum algebra, geometric crystals and soliton cellular automata. Directions (iii), (ii), the dynamical part of (v) and parts of (vi) have since been settled - notably through the 3D boundary consistency condition for quad-graph systems with boundary, and through the algebraic theory of braces (a brace is a set carrying two group structures, one of them abelian, twisted together, whose skew version produces set-theoretic Yang-Baxter solutions). What survives is stated in the Problem Statement below.

Problem Statement

The problem is the unresolved open core of the directions posed in the conclusions of Caudrelier-Crampe-Zhang (2013), after the parts settled by subsequent literature. It comprises three linked components, jointly completing the conceptual foundations of set-theoretical reflection maps - the geometric, physical and algebraic facets of the same classified objects:

(A) Geometric reinterpretation of the classification. Provide a geometric characterization of the reflection maps that Caudrelier-Crampe-Zhang obtained by folding the quadrirational Yang-Baxter maps - in the spirit of Adler-Bobenko-Suris's use of pencils of conics, which gave the bulk classification its geometric meaning - that explains conceptually which reflection maps exist for each quadrirational family covered by that classification (the families FIF_I-FVF_V and the symmetry-related families HIH_I-HVH_V, which carry the same reflection maps, the degenerate maps, and the involutive and non-involutive cases, including the FVF_V family for which the folding classification found no nontrivial solution), ideally with a transparent completeness argument.

(B) Reflection maps as discrete soliton-boundary scattering. Determine whether the interpretation of a reflection map as encoding the integrable interaction between a soliton and a boundary - established for the continuum vector nonlinear Schrödinger equation on the half-line - extends to fully discrete integrable systems. Concretely: for an Adler-Bobenko-Suris quad-graph equation on a half-plane with an integrable boundary satisfying the 3D boundary consistency condition (as classified by Sun-Zhang), or for a discrete vector soliton system such as the integrable discrete Manakov system (a vector Ablowitz-Ladik equation, i.e. the space-discretized Manakov system) on a half-lattice with an integrable boundary, prove - or refute - that the reflection of discrete solitons off the boundary is encoded by a reflection map acting on the discrete soliton data and satisfying the set-theoretical reflection equation. Constructing an inverse scattering transform with boundary (the method that reconstructs solutions from scattering data, here on a half-plane or half-lattice) is one possible route toward this determination, but the question is not restricted to any particular method.

(C) Set-theoretical quadratic boundary algebras. Formulate the set-theoretical analog of the quadratic algebras of Freidel and Maillet: a system of functional equations among four set-theoretical maps generalizing the set-theoretical reflection equation in the same way the four-matrix quadratic algebras generalize the quantum reflection equation. Establish that the formulation is sound, and either construct nontrivial families of solutions or prove that the set-theoretical four-object framework degenerates (for instance, that it forces involutive or trivial maps).

The three components are the surviving residue of one research programme, each closing a different facet (geometric, physical, algebraic) of the same classified objects and each corresponding to one of the directions left open in the conclusions of Caudrelier-Crampe-Zhang (2013). Each component can be attacked and resolved on its own, so a determination of any single component is a partial resolution of the problem, and a complete resolution requires all three. Within component (B), the two stated settings - an Adler-Bobenko-Suris quad-graph equation on a half-plane, or a discrete vector soliton system on a half-lattice - are alternative concretizations of the same question, and establishing the determination in either one settles the component.

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