[Paper Review] Constructing Initial Value Spaces of Lattice Equations

Nalini Joshi, Sarah Lobb+1 more|arXiv (Cornell University)|Jul 17, 2018

Nonlinear Waves and Solitons11 references3 citations

TL;DR

This paper constructs initial value spaces for integrable lattice equations from the ABS list by resolving singularities through algebraic geometry techniques, leading to new Miura transformations that map multiple ABS equations—including H3, Q1, Q3, and A1—into a single lattice equation (3.1). The framework enables reductions to discrete Painlevé equations, revealing new connections and symmetries among integrable systems.

ABSTRACT

In this paper, we examine the space of initial values for integrable lattice equations, which are lattice equations classified by Adler {\em et al} (2003), known as ABS equations. By considering the map which iterates the solution along particular directions on the lattice, we perform resolutions of singularities for several examples of ABS equations for the first time. Our geometric observations lead to new Miura transformations and reductions to ordinary difference equations.

Motivation & Objective

• To extend the concept of initial value spaces—previously developed for discrete Painlevé equations—to higher-dimensional integrable partial difference equations.
• To resolve singularities in the initial value spaces of ABS lattice equations using algebraic geometry, particularly blow-ups of points and lines.
• To uncover new geometric and algebraic structures linking different members of the ABS list through Miura transformations.
• To derive reductions of the resolved lattice equations to ordinary difference equations, specifically QRT maps and discrete Painlevé equations.
• To establish a geometric framework that reveals hidden symmetries and transformations among integrable lattice systems.

Proposed method

• Embedding ABS equations in projective space $\mathbb{P}^3$ to compactify and regularize the initial value space.
• Performing resolution of singularities via sequential blow-ups of four lines and four points for the H3 $\delta=0$ case.
• Using the resolved space to define new variables $u_{l,m}$ and $v_{l,m}$ that transform the original lattice equations into a unified form (equation 3.1).
• Applying Miura transformations to map multiple ABS equations (H3, Q1, Q3, A1) to the same lattice equation (3.1), revealing hidden equivalences.
• Reducing the unified equation (3.1) to a QRT map via periodicity assumptions, leading to a discrete Painlevé equation of surface type $A_3^{(1)}$.
• Utilizing scaling symmetries to reduce the number of parameters in the unified equation to a single parameter $\gamma$, yielding equation (3.21).

Experimental results

Research questions

• RQ1 How can the initial value space of a 3D integrable lattice equation be regularized when singularities arise?
• RQ2 What geometric transformations (e.g., Miura maps) emerge naturally from the resolution process of initial value spaces?
• RQ3 Can multiple ABS equations be unified under a single lattice equation via a common transformation framework?
• RQ4 What reductions to ordinary difference equations (e.g., QRT maps) can be derived from the resolved lattice systems?
• RQ5 How do scaling symmetries and periodic reductions connect the unified lattice equation to known discrete Painlevé equations?

Key findings

• The resolution of singularities for the H3 $\delta=0$ equation is achieved by blowing up four lines and four points, fully regularizing the initial value space.
• A new Miura transformation maps the ABS equations H3 $\delta=0$, Q1 $\delta$, Q3 $\delta=0$, and A1 $\delta=0$ to a single lattice equation (3.1), establishing a unifying transformation.
• Equation (3.1) admits a reduction to a QRT map of surface type $A_3^{(1)}$, given by equation (3.22), which corresponds to a discrete Painlevé equation.
• The unified equation (3.1) can be reduced to a single-parameter form (3.21) via scaling symmetry, simplifying the system to one essential parameter $\gamma$.
• The composition of the Miura transformation and the reduction yields new reductions of the original ABS equations to discrete Painlevé equations.
• The framework reveals deep geometric connections between integrable lattice equations and suggests new paths for constructing Lax pairs and analyzing algebraic entropy.