A systematic approach to reductions of type-Q ABS equations

2013/07/12 by Mike Hay, Phil Howes, Hay, Mike +5 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #Polynomial and algebraic computation #nlin.SI

paper · pdf · doi:10.48550/arxiv.1307.3390

23 pages, 2 figures

openalex publication_date 2013/07/12 · arxiv created 2015/01/11 · arxiv updated 2015/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 A₁⁽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.

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• Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables

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