[Paper Review] ${\mathbb{Z}}_N$ graded discrete Lax pairs and discrete integrable systems

Allan P. Fordy, Pavlos Xenitidis|arXiv (Cornell University)|Nov 22, 2014

Nonlinear Waves and Solitons16 references3 citations

TL;DR

This paper introduces a systematic classification of ${\mathbb{Z}}_N$-graded discrete Lax pairs using $N\times N$ matrices linear in the spectral parameter, enabling the construction of integrable discrete systems with rich algebraic structure. The key contribution is a unified framework that generalizes known integrable equations (e.g., discrete KdV, Boussinesq, Toda lattice) and yields new systems for $N \geq 3$, including novel nine-point and six-point equations, while establishing connections to Bäcklund transformations and continuous symmetries via master symmetries and nonlocal symmetries.

ABSTRACT

We introduce a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs, with $N imes N$ matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated discrete integrable systems. We present two potential forms and completely classify the generic case. Many well known examples belong to our scheme for $N=2$, so many of our systems may be regarded as generalisations of these. Even at $N=3$, several new integrable systems arise. Many of our equations are mutually compatible, so can be used together to form "coloured" lattices. We also present continuous isospectral deformations of our Lax pairs, giving compatible differential-difference systems, which play the role of continuous symmetries of our discrete systems. We present master symmetries and a recursive formulae for their respective hierarchies, for the generic case. We present two nonlocal symmetries of our discrete systems, which have a natural representation in terms of the potential forms. These give rise to the two-dimensional Toda lattice, with our nonlinear symmetries being the Bäcklund transformations and our discrete system being the nonlinear superposition formula (for the generic case).

Motivation & Objective

• To develop a systematic classification scheme for discrete integrable systems using ${\mathbb{Z}}_N$-graded $N\times N$ Lax pairs linear in the spectral parameter.
• To generalize well-known integrable equations (e.g., discrete KdV, modified KdV, Boussinesq) to multi-component systems via this framework.
• To identify and analyze continuous isospectral deformations as local symmetries and construct master symmetries for generating hierarchies of symmetries.
• To establish connections between the discrete systems and the 2D Toda lattice through nonlocal symmetries, identifying them as Bäcklund transformations and nonlinear superposition formulas.
• To explore compatibility of systems for constructing consistent 2D and 3D lattices with non-standard initial value problems.

Proposed method

• Introduce ${\mathbb{Z}}_N$-graded matrix structures with level decomposition to define Lax pairs and associated discrete systems.
• Define two potential forms—quotient and additive—for the generic case, enabling dual descriptions of the same system.
• Use equivalence relations and reduction procedures to classify inequivalent systems and identify coprime and degenerate cases.
• Derive isospectral deformations of the Lax matrices $L$ and $M$ to generate differential-difference equations, interpreted as continuous symmetries of the discrete system.
• Construct master symmetries via non-autonomous flows that generate commuting hierarchies of symmetries in the generic coprime case.
• Identify two nonlocal symmetries (x- and y-flows) that act as Bäcklund transformations for the 2D Toda lattice in potential form, with explicit matrix representations.

Experimental results

Research questions

• RQ1 How can ${\mathbb{Z}}_N$-graded $N\times N$ Lax pairs linear in the spectral parameter be systematically classified to generate discrete integrable systems?
• RQ2 What are the structural and algebraic properties of the resulting discrete systems, particularly for $N \geq 3$, and how do they generalize known integrable equations?
• RQ3 How do continuous isospectral deformations of the Lax matrices relate to local symmetries and hierarchies of differential-difference equations?
• RQ4 In what way do the nonlocal symmetries of the discrete system correspond to Bäcklund transformations of the 2D Toda lattice?
• RQ5 What conditions ensure compatibility between different discrete systems to form consistent 2D and 3D lattices with non-standard initial value problems?

Key findings

• The framework yields new integrable systems for $N=3$, including a nine-point scalar equation derived from a quotient potential form (equation 3.12) and a six-point scalar equation from additive potentials (equation 3.13).
• For $N=2$, the scheme recovers well-known systems such as the discrete potential KdV, modified KdV, and discrete sine-Gordon equations.
• The degenerate case leads to multi-component generalizations of Hirota’s KdV equation (equation 3.53), including a novel $3\times3$ system.
• The master symmetry construction generates commuting hierarchies of symmetries in the generic coprime case, though commutativity is not proven but not contradicted by examples.
• Nonlocal symmetries in the quotient potential form act as Bäcklund transformations for the 2D Toda lattice, with three distinct transformations arising in the $3$-component case ($\ell_i - k_i = 1$).
• The discrete system in the generic case corresponds to the nonlinear superposition formula of the 2D Toda lattice, confirming its integrability and unifying structure.