[Paper Review] Systems of difference equations on a vector valued function that admit 3D space of scalar potentials

Pavlos Kassotakis, Maciej Nieszporski|arXiv (Cornell University)|Aug 5, 2019

Nonlinear Waves and Solitons16 references4 citations

TL;DR

This paper investigates integrable systems of difference equations on vector-valued functions that admit a 3D space of scalar potentials, deriving invariants with separated variables for specific involutive maps. By linking these invariants to potential functions on lattice vertices, the authors reformulate edge-based bond systems into vertex-based models, recovering known equations like the lattice potential KdV and discovering new non-single-valued difference relations resolved via potential structures.

ABSTRACT

For some involutive maps $Φ:{\mathbb C}P^1 imes {\mathbb C}P^1 o {\mathbb C}P^1 imes {\mathbb C}P^1$ we find all invariants with separated variables. We investigate a link of the maps and their invariants with separated variables to discrete integrable systems. Maps correspond to integrable systems on edges (bond systems), while their invariants with separated variables yields potentials of the bond systems, that allows us to rewrite the integrable sytems as models on vertices. Among the latter ones one can find well known integrable difference equations as well as difference relations, which in contrast to the equations give non-single-valued evolution of the dependent variable. However, the non-single-valuedness can be resolved by the link with the bond system.

Motivation & Objective

• To identify all invariants with separated variables for a class of involutive maps on $\mathbb{C}P^1 \times \mathbb{C}P^1$.
• To establish a systematic link between such invariants and scalar potentials that transform bond systems (on edges) into vertex-based systems.
• To classify and analyze the resulting vertex equations and relations, including non-single-valued evolutions resolved through potential structures.
• To explore the role of non-auto Bäcklund transformations in connecting solutions of different integrable equations from the ABS list.
• To provide a complete characterization of the 3D affine space of alternating invariants with separated variables for the studied maps.

Proposed method

• Derive and classify all alternating invariants with separated variables for involutive maps of the form $U = v + \frac{p-q}{u-v}$, $V = u + \frac{p-q}{u-v}$, using algebraic and polynomial techniques.
• Construct scalar potentials $\psi_{m,n}$ from invariants satisfying $I(U,V) = -I(u,v)$, enabling the transformation of edge-based systems into vertex-based models.
• Re-express the original bond system as a lattice potential KdV equation: $(\psi_{m+1,n+1} - \psi_{m,n})(\psi_{m+1,n} - \psi_{m,n+1}) = p_{m_1} - q_{n_2}$.
• Systematically analyze the resulting vertex equations, including multilinear (e.g., $A2$, $H3^{\pm1}$) and higher-degree relations (e.g., $H2^*$, $G2$, $F1$) arising from different parameter choices.
• Construct non-auto Bäcklund transformations mapping solutions of one equation (e.g., $A2$) to solutions of another via rational functions of the form $y_i = \frac{s^i(x,x_i)y + u^i(x,x_i)}{t^i(x,x_i)y + v^i(x,x_i)}$.
• Use the consistent-around-the-cube property and multi-quadratic/multi-quartic relations to verify integrability and structural consistency across families.

Experimental results

Research questions

• RQ1 What is the complete set of alternating invariants with separated variables for the given class of involutive maps?
• RQ2 How can invariants with separated variables be used to construct scalar potentials that reformulate edge-based difference systems into vertex-based models?
• RQ3 What types of vertex equations and non-single-valued difference relations emerge from different parameter choices in the invariant family?
• RQ4 How do non-auto Bäcklund transformations connect solutions of different integrable equations in the ABS list?
• RQ5 What is the role of the 3D space of scalar potentials in resolving non-single-valued evolutions in difference systems?

Key findings

• The alternating invariants with separated variables for the map $\Phi$ form a 3-dimensional affine space, fully parameterized by constants $a$, $b$, $c$, $d$ in the expression $I(u,v) = a(u-v) + b(u^2 + p - v^2 - q) + c(u^3 + 3pu - v^3 - 3qv) + d$.
• The lattice potential KdV equation emerges as a vertex model when the invariant corresponds to $a=1$, $b=c=d=0$, yielding $u_{m_1,n_1+1} + u_{m_1,n_1} = v_{m_2+1,n_2} + v_{m_2,n_2}$ and $\psi$-based evolution.
• Non-single-valued difference relations arise from certain parameter choices (e.g., $b=1$, $a=c=d=0$), but their non-uniqueness is resolved via the underlying bond system and potential structure.
• The system $I_I$ is spanned by three $A2$ equations, which decompose into three versions of multi-quadratic $A2^*$ relations, indicating a deep algebraic structure.
• The systems $I_{II}$ and $I_{III}$ exhibit a quark-like structure resembling neutron-proton systems, with $H3^{\pm1}$ and $H3^0$ relations forming their core.
• Nonlinear relations such as $H2^*$ (multi-quadratic), $G2$ (multi-quartic), and $F1$ (degree-nine) appear in the bottom models $I_{IV}$ and $I_V$, showing the richness of the potential space.