Weak Lax pairs for lattice equations

We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and B"acklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional Yang-Baxter equation.

💡 Research Summary

The paper “Weak Lax pairs for lattice equations” investigates the relationships among three cornerstone concepts in the theory of integrable discrete systems: three‑dimensional consistency (often called Consistency‑Around‑a‑Cube, CAC), Lax pairs, and Bäcklund transformations (BT). While earlier work suggested that these notions are essentially equivalent—CAC guaranteeing the existence of a Lax pair and a BT—the authors demonstrate that the equivalence is not strict. In particular, they introduce the notion of a “weak” Lax pair: a pair of matrices that satisfy the zero‑curvature condition (ZCC) but do not uniquely determine the underlying two‑dimensional lattice equation, or may give rise to several admissible equations.

The exposition begins with a concise review of two‑dimensional lattice equations defined on a square lattice by a multiaffine relation Q(x, x₁, x₂, x₁₂)=0, where the indices denote shifts in the two lattice directions. By placing copies of this relation on the six faces of a cube, the authors recall the definition of CAC: the three possible ways of computing the value at the opposite corner x₁₂₃ must coincide. CAC is known to produce both a Lax representation and a BT, but the paper asks how far this implication can be pushed.

Section 3 develops the construction of a Lax pair from the side equations of the cube. By solving the side equations for the “spectral” variables x₃, x₁₃, x₂₃, x₁₂₃ and introducing homogeneous coordinates (f,g) with x₃ = f/g, the authors define a two‑component wave function ψ = (f,g)ᵀ. The side equations become linear relations ψ₁ = L ψ and ψ₂ = M ψ, where L and M are 2 × 2 matrices whose entries are affine functions of the corner variables. The zero‑curvature condition L(x₁₂,x₂) M(x₂,x) = M(x₁₂,x₁) L(x₁,x) yields three scalar equations. In genuinely integrable cases these three equations share a common factor, which is precisely the original two‑dimensional equation (the “bottom” equation). However, the authors point out that the common factor may be absent, may factor further, or may be non‑unique, leading to a weak Lax pair.

The BT construction is presented in parallel. By sequentially eliminating x₁₃, x₂₃, and x₁₂₃ from the side equations, the right‑hand side equation reduces to a quadratic polynomial R₂ x² + R₁ x + R₀ = 0, where the coefficients Rᵢ are polynomials in the four base variables (x, x₁, x₂, x₁₂). The greatest common divisor (GCD) of R₂, R₁, R₀ provides the bottom equation. The authors show that the matrices U = M L and V = L M encode the same information: the relations U·V⁻¹ = 0 are equivalent to the vanishing of the Rᵢ. Thus, the Lax‑pair and BT approaches are algebraically equivalent, but only when the GCD is non‑trivial.

Section 4 supplies concrete examples that illustrate the various possibilities.

1. Linear side equations – When each side equation is linear (e.g., x₁₃ − x₁ − x₃ + x = 0), neither a non‑trivial Lax pair nor a BT emerges; the ZCC is automatically satisfied. Nevertheless, CAC can still be imposed, leading to a mixed linear‑KdV type bottom equation, showing that CAC alone does not guarantee a meaningful Lax pair.

2. H1 (the lattice potential KdV) – The classic H1 equation (x₁ − x₂)(x − x₁₂) − p + q = 0 satisfies CAC, yields a standard Lax pair L =