[Paper Review] A systematic approach to reductions of type-Q ABS equations

Mike Hay, Phil Howes+2 more|arXiv (Cornell University)|Jul 12, 2013

Nonlinear Waves and Solitons24 references3 citations

TL;DR

This paper presents a systematic method for reducing type-Q ABS equations (Q1, Q2, Q3) via Möbius-type periodic constraints, demonstrating that most reductions yield linearisable mappings through birational transformations derived from the geometry of initial value spaces. The key contribution is the discovery of a $q$-Painlevé equation on the $A_1^{(1)}$ surface arising from a deautonomised reduction of Q3, confirmed via singularity confinement and degree growth analysis.

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.

Motivation & Objective

• To develop a systematic framework for periodic reductions of Q1, Q2, and Q3 equations from the ABS list using Möbius-type constraints.
• To determine which reductions lead to integrable mappings, particularly focusing on linearisability via algebraic entropy and space of initial conditions.
• To identify and construct explicit linearising birational transformations for reductions that exhibit linear degree growth.
• To explore the existence of non-linearisable reductions, especially for Q3, and to analyze their singularity structure.
• To demonstrate the emergence of a $q$-Painlevé equation on the $A_1^{(1)}$ surface via deautonomisation using singularity confinement.

Proposed method

• Apply reductions of the form $u(k,l+1) = m(u(k+1,l))$, where $m$ is a Möbius transformation, to Q1, Q2, and Q3 equations.
• Use algebraic entropy (degree growth of iterates) to classify integrability: linear growth implies linearisability.
• Construct the space of initial conditions via blow-ups of $\mathbb{P}^2$ at base points, enabling geometric analysis of the mapping.
• For linearisable cases, derive explicit birational transformations that convert the system into cascaded Riccati maps.
• For the non-linearisable Q3 reduction, apply singularity confinement to deautonomise the system and identify the Painlevé surface.
• Analyze the action of the mapping on the Picard group to compute degree growth and confirm unconfined singularities in non-linearisable cases.

Experimental results

Research questions

• RQ1 Which Möbius-type reductions of Q1, Q2, and Q3 ABS equations yield linearisable mappings?
• RQ2 Can the space of initial conditions be used to explicitly construct linearising transformations for these reductions?
• RQ3 What is the nature of the singularity structure in non-linearisable reductions, particularly for Q3?
• RQ4 Does a deautonomised reduction of Q3 give rise to a known discrete Painlevé equation in Sakai’s classification?
• RQ5 Can the $A_1^{(1)}$ surface in Sakai’s classification be realized as the geometric base for a $q$-Painlevé equation derived from an ABS equation?

Key findings

• All reductions of Q1 and Q2 equations with Möbius-type constraints are linearisable, as evidenced by linear degree growth.
• Explicit birational transformations are constructed that linearise the reductions into cascaded Riccati maps, enabling algorithmic solution.
• For Q3, three reductions are found to be linearisable, while one specific periodic reduction exhibits quadratic degree growth.
• The quadratic growth reduction of Q3 is deautonomised via singularity confinement, yielding a $q$-Painlevé equation.
• The resulting $q$-Painlevé equation is geometrically realized on the $A_1^{(1)}$ surface in Sakai’s classification of Painlevé systems.
• The mapping exhibits unconfined singularities requiring an infinite sequence of blow-ups, confirmed by the Picard group action showing unbounded growth in exceptional divisor coefficients.