A systematic approach to reductions of type-Q ABS equations

Abstract

Abstract. We present a class of reductions of Möbius type for the lattice equations known as Q1, Q2, and Q3 from the ABS list. The deautonomised form of one particular reduction of Q3 is shown to exist on the A (1) 1 surface which belongs to the multiplicative type of rational surfaces in Sakai's classification of Painlevé systems. Using the growth of degrees of iterates, all other mappings that result from the class of reductions considered here are shown to be linearisable. Any possible linearisations are calculated explicitly by constructing a birational transformation defined by invariant curves in the blown up space of initial values for each reduction.

Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation

“…This work on the symmetries of reduced equations applies a wide class of periodic reductions that have appeared in the literature. It is not entirely clear at this point how, or even whether, this procedure may be applied to the so-called twisted reductions explored by recent work [13,38]. This work suggests that the symmetry group of a (s 1 , s 2 )-reduction is at least a lattice of dimension s 1 + s 2 , hence, we suspect it would take a (4, 4)-reduction of Q4 to be able to be identified with the full parameter version of the elliptic Painlevé equation.…”

“…Finding explicit solutions to integrable partial differential equations in terms of solutions of ordinary differential equations, such as the Painlevé equations, is a topic that is of interest to many researchers [1,7,10,28]. Finding explicit solutions to the discrete analogues of integrable partial differential equations, integrable lattice equations, in terms of known ordinary difference equations, such as the discrete Painlevé equations, has recently been a hot topic [12,13,14,37,38,39,45].…”

Lax pairs of discrete Painlevé equations: (A2+A1)(1)case

“…We note that, in general, a hypercube is said to be multidimensionally consistent, if all cubes contained in the hypercube are 3D consistent (see property (2) above). Reductions of such ABS equations to ordinary difference equations have been found through several approaches [5,[28][29][30][31][32][33][34][35]. Our geometric-reduction method [1][2][3]36] has shown how to obtain discrete Painlevé equations by studying geometric connections between these and ABS equations.…”

Geometric reductions of ABS equations on ann-cube to discrete Painlevé systems

“…Previous studies in the literature have performed reductions of such equations via methods suited to specific examples [17,9,13,10,8,21,11,22]. In particular, the identification of the reduced system has been mainly achieved by comparing or transforming it to known forms of the discrete Painlevé equations.…”