Multi-Component Extension of CAC Systems

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 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.

Discrete Lax pairs and hierarchies of integrable difference systems

“…and the resulting birational system will be called companion system. Now we are ready to prove that ( 15) is quadrirational provided that the centrality assumptions (19) are imposed. From ( 12) we obtain…”

“…The hierarchy of difference systems ( 15) is birational. If the centrality assumptions (19) are imposed, then (15) is quadrirational.…”

“…. , φ N }, {φ }, that is a reminiscence of the associated quadrirational difference system in edge variables (15), (19).…”

“…In the discrete scheme, results on multi-component partial difference systems, or scalar partial difference equations defined on higher order stencils, are rather sparse [8,9,10,11,12,13,14,15,16,17,18,19,20]. Moreover, there are just a few results on hierarchies of integrable partial difference equations, for example discrete analogues of the Gel'fand-Dikii hierarchy, of the modified and the Schwarzian Gel'fand-Dikii hierarchies have been introduced in the past.…”

Local and nonlocal complex discrete and semi-discrete sine-Gordon equations and solutions

“…There was a little work on the nonlocal discrete integrable systems. In [19], Zhang et al investigated two types of nonlocal discrete integrable equations, which were constructed as reductions of 2-component Adler-Bobenko-Suris systems [20]. In particular, they showed that the 2[0, 1] Adler-Bobenko-Suris system allows reverse-n nonlocal reduction and the 2 [1,1] Adler-Bobenko-Suris system admits reverse-(n, m) nonlocal reduction.…”