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

TL;DR

This paper extends the concept of three-dimensional consistency to a 3x3x3 cube for lattice Boussinesq equations and introduces a new lattice BSQ-Q3 system as an analogue of the Q3 equation, advancing integrability theory.

Contribution

It demonstrates the extension of 3D consistency to non-quadrilateral lattices and constructs a novel lattice BSQ-Q3 system related to the ABS classification.

Findings

Extended 3D consistency to a 3x3x3 cube for lattice BSQ equations.

Constructed a new lattice BSQ-Q3 system as an analogue of Q3 in ABS classification.

Derived a PGL_3-invariant generalization of the Schwarzian BSQ equation.

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($\delta$) equation in the Adler-Bobenko-Suris (ABS) classification. The construction relies on a gauge transformation between Lax pairs of lpBSQ with the parameter $\delta$ 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.