Reflection groups and discrete integrable systems

Kamp

2020

SIGMA

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.

Section: N -Component Q-painlevé III Equationmentioning

confidence: 99%

Multi-Component Extension of CAC Systems

Kamp

2020

SIGMA

J. Phys. A: Math. Theor.

“…π (34) .k = 1 k , π (34) .b 1 = −b 1 /k 2 , π (34) .b 3 = −k 2 b 4 , π (34) .b 2 = −b 2 /k 2 , π (34) .b 4 = −k 2 b 3 , π (14) :…”

Section: -Surfacesunclassified

“…π (14) .k = k/(k 2 − 1) 1/2 , π (14) .b 1 = b 1 /(k 2 − 1) 2 , π (14) .b 2 = b 4 , π (14) .b 3 = (k 2 − 1) 2 b 3 , π (14) .…”

Section: -Surfacesunclassified

Full-parameter discrete Painlevé systems from non-translational Cremona isometries

Stokes¹

2018

Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai's list. For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries. Translation elements of this symmetry group define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-type through projective reduction, these have fewer than the maximal number of free parameters corresponding to their surface type. We show that difference equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlevé equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature, which we generalise to a full-parameter integrable difference equation, given by the Cremona action of a non-translation element of the extended affine Weyl group W (D (1) 4 ) on a family of generic D (1) 4 -surfaces.2

“…There are close relations, cf. [25,29], between non-autonomous ABS lattice equations and discrete Painlevé equations exploing the affine Weyl group. A particular example is provided by a non-autonomous version of the lpmKdV equation (2.30) which can be reduced to a q-Painlevé III equation, by performing a periodic reduction.…”

Section: N -Component Q-painlevé III Equationmentioning

confidence: 99%

Multi-Component Extension of CAC Systems

Zhang,

van der Kamp,

Zhang

2019

Preprint

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 multicomponent discrete Painlevé equations can also be derived in the context, and the approach extends to N -component generalizations of higher dimensional lattice equations.