On consistency around a 3 × 3× 3 cube and Q3 analogue of the lattice Boussinesq equation

2026/01/09 by Pengyu Sun, Cheng Zhang, Frank Nijhoff · 1 voice
Physics and Astronomy · #nlin.SI #math-ph

paper · pdf

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× 3× 3 cube\textemdash a cubic sublattice consisting of 27 vertices. This extends the standard notion of three-dimensional consistency (defined on an elementary 2× 2× 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 GL₃ action. In a degeneration form, the system yields a PGL₃-invariant integrable lattice equation that generalises the PGL₂-invariant Schwarzian BSQ equation.

Citations

• Discrete Boussinesq-type equations
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• A higher-rank version of the Q3 equation
• Classification of integrable discrete equations of octahedron type
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• Soliton solutions for ABS lattice equations: I. Cauchy matrix approach
• The Pentagram map: a discrete integrable system
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• Backlund transformations for integrable lattice equations
• Soliton solutions for Q3
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• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• The Boussinesq integrable system. Compatible lattice and continuum structures
• Classification of integrable equations on quad-graphs. The consistency approach
• Lax pair for the Adler (lattice Krichever–Novikov) system
• Integrable systems on quad-graphs
• Menelausapos theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
• The Discrete and Continuous Painleve VI Hierarchy and the Garnier Systems
• Bäcklund transformation for the Krichever-Novikov equation

Discussions

• Pengyu Sun, Cheng Zhang, Frank Nijhoff: On consistency around a $3 \times 3\times 3$ cube and Q3 analogue of the lattice Boussinesq equation https://arxiv.org/abs/2601.05565 https://arxiv.org/pdf/2601 [bsky, 0 points, 0 comments]

Related

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• Boussinesq-like multi-component lattice equations and multi-dimensional consistency
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• On decomposition of the ABS lattice equations and related Bäcklund transformations