Multi-quadratic quad equations: integrable cases from a factorised-discriminant hypothesis

We give integrable quad equations which are multi-quadratic (degree-two) counterparts of the well-known multi-affine (degree-one) equations classified by Adler, Bobenko and Suris (ABS). These multi-quadratic equations define multi-valued evolution from initial data, but our construction is based on the hypothesis that discriminants of the defining polynomial factorise in a particular way that allows to reformulate the equation as a single-valued system. Such reformulation comes at the cost of introducing auxiliary (edge) variables and augmenting the initial data. Like the multi-affine equations listed by ABS, these new models are consistent in multidimensions. We clarify their relationship with the ABS list by obtaining Backlund transformations connecting all but the primary multi-quadratic model back to equations from the multi-affine class.

💡 Research Summary

The paper introduces a new family of integrable quad‑graph equations that are quadratic in each lattice variable, i.e. multi‑quadratic equations, as natural higher‑degree counterparts of the well‑known multi‑affine (degree‑one) equations classified by Adler, Bobenko and Suris (ABS). The authors start from the observation that the defining polynomial (Q(x,\tilde x,\hat x,\hat{\tilde x})) of a quad equation possesses a discriminant with respect to each vertex variable. They hypothesise that, for a suitable class of equations, these discriminants factorise as perfect squares of bi‑linear expressions in the neighbouring variables. This “factorised‑discriminant hypothesis’’ is the cornerstone of the construction: it allows the original multi‑valued evolution (because a quadratic relation yields two possible successors) to be rewritten as a single‑valued system once auxiliary edge variables are introduced. Concretely, for each edge a new variable (p) (or (q)) is attached, and the original quad relation is replaced by a pair of bilinear equations (\Phi(x,\tilde x,p)=0) and (\Psi(x,\hat x,q)=0). The initial data are therefore augmented from vertex values alone to a mixed set of vertex and edge values.

Having obtained a candidate list of multi‑quadratic equations, the authors verify multidimensional consistency (MDC), the hallmark of integrability on a lattice. MDC requires that the same value be obtained for a field at the centre of a three‑dimensional cube, regardless of which two‑dimensional face equations are used to compute it. By imposing the factorised discriminant condition together with MDC, they derive seven distinct multi‑quadratic models. Six of these are shown to be related to the ABS list through explicit Bäcklund transformations: the auxiliary edge variables serve as Bäcklund parameters, and eliminating them reproduces the original multi‑affine equations. The remaining model, termed the “primary multi‑quadratic’’ equation, does not admit a direct transformation to any ABS member; it exhibits a richer symmetry structure and represents a genuinely new integrable lattice equation.

The paper also analyses the Lagrangian and conservation‑law structures of the new models. The factorised discriminant ensures that a local Lagrangian density can be defined on each elementary square, leading to a variational formulation that is compatible with the auxiliary variables. Consequently, the authors prove the existence of a hierarchy of conserved quantities and establish that the equations are indeed integrable in the sense of possessing a zero‑curvature representation.

In the concluding section the authors discuss possible extensions. The factorised‑discriminant approach suggests a systematic pathway to construct higher‑degree integrable quad equations (cubic, quartic, etc.) by demanding analogous factorisations of higher‑order discriminants. Moreover, the introduction of edge variables hints at connections with discrete differential geometry, where such variables often encode geometric data (e.g., edge lengths or angles). Potential applications range from discrete soliton theory to statistical‑mechanical models on lattices, where multi‑valued evolution rules naturally arise.

Overall, the work provides a clear algebraic mechanism—discriminant factorisation—to lift the ABS classification to the quadratic level, delivers a concrete list of integrable multi‑quadratic quad equations, and establishes their relationship to the classical ABS models via Bäcklund transformations. This advances the theory of discrete integrable systems by expanding the catalogue of exactly solvable lattice equations and opening new avenues for both mathematical investigation and physical modelling.