Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 060, 30 pages      arXiv:1912.00713

Multi-Component Extension of CAC Systems

Dan-Da Zhang a, Peter H. van der Kamp b and Da-Jun Zhang c
a) School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China
b) Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
c) Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China

Received December 07, 2019, in final form June 14, 2020; Published online July 01, 2020

In this paper an approach to generate multi-dimensionally consistent $N$-component systems is proposed. The approach starts from scalar multi-dimensionally consistent quadrilateral systems and makes use of the cyclic group. The obtained $N$-component systems inherit integrable features such as Bäcklund transformations and Lax pairs, and exhibit interesting aspects, such as nonlocal reductions. Higher order single component lattice equations (on larger stencils) and multi-component discrete Painlevé equations can also be derived in the context, and the approach extends to $N$-component generalizations of higher dimensional lattice equations.

Key words: lattice equations; consistency around the cube; cyclic group; multi-component; Lax pair; Bäcklund transformation; nonlocal.

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