On consistency around a $3 \times 3\times 3$ cube and Q3 analogue of the lattice Boussinesq equation

TL;DR

The paper tackles extending integrable lattice BSQ-type equations beyond the KdV/ABS framework by proving a consistency-around-a-$3\times 3\times 3$ cube for the nine-point lpBSQ equation and by constructing a lattice BSQ-Q3 system that serves as the BSQ analogue of Q3($\delta$). The BSQ-Q3 system is derived via a discrete gauge between two lpBSQ Lax pairs with distinct spectral parameters, introducing a GL$_3$-induced parameter $\delta$ and yielding a 3D-consistent, three-component lattice equation. A degeneration to a $PGL_3$-invariant autonomous system is also established, linking to a BSQ Schwarzian-type equation. These results expand the catalog of integrable lattice equations to higher-rank, non-quadrilateral settings and suggest pathways toward a higher-rank ABS-like classification and geometric interpretations. Future work includes obtaining exact solutions, exploring further degenerations to BSQ analogues of Q2 and Q1, and clarifying the underlying projective- and Grassmannian-geometric structures.

Abstract

In this paper, we present two new aspects of lattice Boussinesq (BSQ) equations. First, we show that the lattice potential BSQ (lpBSQ) equation defined on a nine-point square lattice admits a natural extension of three-dimensional consistency to a $3\times 3\times 3$ cube\textemdash a cubic sublattice consisting of $27$ vertices. This extends the standard notion of three-dimensional consistency (defined on an elementary $2\times 2\times 2$ vertex cube for quadrilateral equations) to the non-quadrilateral, nine-point setting. Second, we construct a new three-component system which is referred to as the {\em lattice BSQ-Q3 system}, serving as the BSQ analogue of the Q3($δ$) equation in the Adler-Bobenko-Suris (ABS) classification. The construction relies on a gauge transformation between Lax pairs of lpBSQ with the parameter $δ$ arising from a $GL_3$ action. In a degeneration form, the system yields a $PGL_3$-invariant integrable lattice equation that generalises the $PGL_2$-invariant Schwarzian BSQ equation.

Figures (9)

• Figure 1: 9-point configuration of the lattice BSQ type equations.
• Figure 2: 3D consistency of lpKdV: the six faces of an elementary cube are governed by the lpKdV equations. Having values of $(w, \widetilde{w}, \widehat{w}, \overline{w})$ on the black dots as well as of the lattice parameters $(\alpha, \beta, \gamma)$ as initial data, one first computes the values of $(\widehat{\widetilde{w}}, \widetilde{\overline{w}}, \widehat{\overline{w}})$ on the white squares, then has three ways to determine the value on the white diamond. Consistency means that this is well-posed.
• Figure 3: A $3\times 3\times 3$-vertex cube, the discrete coordinates are fixed. Shifts in independent variables $n,m,\ell \in \mathbb Z$ are respectively denoted by $\widetilde{~}$ , $\widehat{~}$ , $\bar{~}$ , and $\alpha, \beta, \gamma$ are the respective lattice parameters.
• Figure 4: The supporting lattices of the equations $C_{\alpha\beta}(w)=0$, $C_{\beta\gamma}(w)=0$ and $C_{\gamma\alpha}(w)=0$
• Figure 5: The L-shaped BSQ equation ${\cal L}_{\gamma\alpha} (w)= 0$ defined on a 3D nine-point lattice in the $3\times 3\times 3$ cube which can be obtained by eliminating $\widetilde{\widehat{\overline{w}}}$ in $C_{\gamma\alpha}(w)=0$ and $\widehat{H}(w) = 0$.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Proposition 3.1
• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• Lemma 3.5
• Remark 3.6
• Remark 3.7
• Theorem 3.8