[Paper Review] Multi-component generalisation of CAC systems

Dan-da Zhang, Peter H. van der Kamp+1 more|arXiv (Cornell University)|Dec 2, 2019

Numerical methods for differential equations4 citations

TL;DR

This paper proposes a method to construct multi-component generalizations of CAC (Consistent Around the Cube) systems using scalar quadrilateral systems and cyclic group symmetry. By leveraging this structure, the approach generates $N$-component integrable lattice equations with inherited Lax pairs, Bäcklund transformations, and nonlocal reductions, extending to higher-order and multi-component discrete Painlevé equations on larger stencils.

ABSTRACT

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 Backlund 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 Painleve equations can also be derived in the context, and the approach extends to $N$-component generalizations of higher dimensional lattice equations.

Motivation & Objective

• To develop a systematic approach for generating multi-component generalizations of integrable lattice equations.
• To preserve key integrability features—such as Lax pairs and Bäcklund transformations—across $N$-component systems.
• To extend the framework to higher-order single-component equations and multi-component discrete Painlevé equations.
• To explore nonlocal reductions within the generalized $N$-component systems.
• To generalize the construction to higher-dimensional lattice equations.

Proposed method

• Start from scalar multi-dimensionally consistent quadrilateral systems as base structures.
• Apply the cyclic group to systematically extend scalar systems into $N$-component systems.
• Use group action to define component interactions while preserving integrability.
• Construct Lax pairs and Bäcklund transformations for the resulting $N$-component systems.
• Derive higher-order equations on larger stencils through recursive application of the method.
• Extend the framework to $N$-component generalizations of higher-dimensional lattice equations.

Experimental results

Research questions

• RQ1 How can scalar multi-dimensionally consistent lattice systems be systematically generalized to $N$-component systems while preserving integrability?
• RQ2 What role does the cyclic group play in ensuring consistency and integrability in multi-component extensions?
• RQ3 Can nonlocal reductions be naturally incorporated into the $N$-component framework?
• RQ4 How do higher-order single-component lattice equations emerge from this construction?
• RQ5 To what extent can the method be generalized to higher-dimensional lattice equations?

Key findings

• The proposed method successfully generates $N$-component systems that inherit Lax pairs and Bäcklund transformations from the original scalar systems.
• Nonlocal reductions are naturally realized within the $N$-component framework, revealing new structural features.
• Higher-order single-component lattice equations on larger stencils are derived as special cases of the general construction.
• Multi-component discrete Painlevé equations are obtained through the same systematic approach.
• The framework is extendable to $N$-component generalizations of higher-dimensional lattice equations.
• The method ensures multi-dimensional consistency across all derived $N$-component systems.