[Paper Review] Discrete Lax pairs and hierarchies of integrable difference systems

Pavlos Kassotakis|arXiv (Cornell University)|Apr 29, 2021

Nonlinear Waves and Solitons55 references4 citations

TL;DR

This paper introduces a family of discrete Lax matrices $ L^{N,k} $ of order $ N \in \mathbb{N} $, indexed by $ k \in \{1, \ldots, N-1\} $, over a division ring to construct hierarchies of integrable non-commutative difference systems in edge and vertex variables. The key contribution is a unified framework that generates known hierarchies—such as non-commutative lattice-modified and lattice-Schwarzian Gel'fand-Dikii systems—as reductions, while also revealing a novel hierarchy for $ k=2 $ that unifies them.

ABSTRACT

We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge variables that in turn lead to hierarchies of integrable difference systems in vertex variables or in a combination of edge and vertex variables. Furthermore, the entries of the Lax matrices are considered as elements of a division ring, so we obtain hierarchies of discrete integrable systems extended in the non-commutative domain.

Motivation & Objective

• To develop a systematic framework for constructing integrable non-commutative difference systems using discrete Lax pairs.
• To generalize scalar and commutative integrable hierarchies to the non-commutative domain via Lax matrices over a division ring.
• To unify known non-commutative hierarchies—such as the lattice-modified and lattice-Schwarzian Gel'fand-Dikii hierarchies—under a single Lax matrix family $ L^{N,k} $.
• To identify new integrable hierarchies, particularly for $ k=2 $, that include previous hierarchies as reductions.

Proposed method

• Introduces a family of $ N \times N $ Lax matrices $ L^{N,k} $ with entries in a division ring, parameterized by $ k \in \{1, \ldots, N-1\} $.
• Derives linear problems involving $ L^{N,k} $ and $ M^{N,k} $, leading to integrable hierarchies in edge and vertex variables through discrete spectral problems.
• Uses strong Lax pair conditions to ensure integrability and constructs hierarchies via recursion relations in the spectral parameter.
• Applies centrality assumptions to recover explicit forms of non-commutative lattice-modified and lattice-Schwarzian Gel'fand-Dikii hierarchies for $ k=1 $.
• Derives explicit equations for the $ k=2 $ case, showing it contains both $ k=1 $ hierarchies as reductions.
• Proposes a generalized linear problem $ \Psi_2 = L^{N,k_1,a} \Psi $, $ \Psi_1 = M^{N,k_2,b} \Psi $, extending the framework to broader classes of systems.

Experimental results

Research questions

• RQ1 How can a family of discrete Lax matrices $ L^{N,k} $ be constructed to generate integrable hierarchies in non-commutative settings?
• RQ2 What is the relationship between the hierarchies generated by $ L^{N,k} $ for different $ k $, and how do they unify known integrable systems?
• RQ3 Can the $ k=2 $ case yield a novel hierarchy that includes both the non-commutative lattice-modified and lattice-Schwarzian Gel'fand-Dikii hierarchies as reductions?
• RQ4 What is the continuous limit of the discrete hierarchies associated with $ L^{N,k} $, particularly for $ k>1 $?
• RQ5 How can the full hierarchy of Yang-Baxter maps be explicitly constructed from the $ L^{N,k} $ Lax matrices?

Key findings

• For $ k=1 $, the Lax matrix $ L^{N,1} $ generates a hierarchy of non-commutative difference systems in edge variables, which reduces to the non-commutative lattice-modified and lattice-Schwarzian Gel'fand-Dikii hierarchies under centrality assumptions.
• The first two members of the Yang-Baxter map hierarchy associated with $ L^{N,1} $ are explicitly constructed: the first is a 4-parameter extension of the $ H_{III}^{A} $ map in the non-commutative domain.
• For $ k=2 $, a new hierarchy of difference systems in edge and vertex variables is derived, which includes both $ k=1 $ hierarchies as reductions, indicating a unifying structure.
• The explicit form of the $ k=3 $ hierarchy in edge variables is provided via a system of six equations involving $ x^{3,i}, y^{3,i}, x^{1,i}, y^{1,i} $, etc., with solutions expressed in terms of potential functions $ \phi^i, \chi^i $.
• The linear problem $ \Psi_2 = L^{N,k_1,a} \Psi $, $ \Psi_1 = M^{N,k_2,b} \Psi $ generalizes the framework and suggests a broader class of integrable systems.
• The paper identifies open problems, including the continuous limit of $ L^{N,k} $ hierarchies for $ k>1 $, and the explicit construction of Yang-Baxter and entwining maps for $ k \geq 2 $.