[Paper Review] Multi-quadratic quad equations: integrable cases from a factorised-discriminant hypothesis

James Atkinson, Maciej Nieszporski|arXiv (Cornell University)|Apr 3, 2012

Nonlinear Waves and Solitons24 references4 citations

TL;DR

This paper introduces a new class of integrable multi-quadratic quad equations by hypothesizing a specific factorization of the discriminant of their defining polynomial, enabling reformulation as single-valued systems via auxiliary edge variables. The key contribution is a systematic list of such equations, with all but one (the multi-quadratic counterpart of Q4) connected to the well-known multi-affine ABS list through Bäcklund transformations, establishing integrability via multidimensional consistency.

ABSTRACT

We give integrable quad equations which are multi-quadratic (degree-two) counterparts of the well-known multi-affine (degree-one) equations classified by Adler, Bobenko and Suris (ABS). These multi-quadratic equations define multi-valued evolution from initial data, but our construction is based on the hypothesis that discriminants of the defining polynomial factorise in a particular way that allows to reformulate the equation as a single-valued system. Such reformulation comes at the cost of introducing auxiliary (edge) variables and augmenting the initial data. Like the multi-affine equations listed by ABS, these new models are consistent in multidimensions. We clarify their relationship with the ABS list by obtaining Backlund transformations connecting all but the primary multi-quadratic model back to equations from the multi-affine class.

Motivation & Objective

• To identify integrable multi-quadratic quad equations beyond the standard multi-affine class.
• To develop a systematic method for constructing such equations using a discriminant factorization hypothesis.
• To reformulate multi-valued quad equations as single-valued systems using auxiliary edge variables.
• To establish connections between the new multi-quadratic models and the established multi-affine ABS equations via Bäcklund transformations.
• To investigate whether the discriminant factorization property alone is sufficient for integrability in this class.

Proposed method

• Propose a factorized-discriminant hypothesis: the discriminant of the defining quad equation polynomial factors into products of edge biquadratics.
• Construct models from biquadratic polynomials associated with lattice edges, mirroring the approach used in the ABS list.
• Introduce auxiliary edge variables to reformulate the original multi-valued system into a single-valued system with augmented initial data.
• Verify multidimensional consistency of the reformulated systems, confirming integrability.
• Use discriminant properties and algebraic relations to derive Bäcklund transformations connecting multi-quadratic models to multi-affine ABS equations.
• Apply transformation theory based on generalized discriminants and asymmetric discriminant characterizations to identify and construct connecting maps.

Experimental results

Research questions

• RQ1 Can the discriminant factorization property alone ensure integrability in multi-quadratic quad equations?
• RQ2 Which multi-quadratic quad equations can be reformulated as single-valued systems using auxiliary edge variables?
• RQ3 Do Bäcklund transformations exist that connect multi-quadratic models to the multi-affine ABS equations?
• RQ4 What is the role of edge biquadratics in the integrability and reformulation of multi-quadratic systems?
• RQ5 Why is the primary multi-quadratic model (counterpart of Q4) the only one not yet connected to the ABS list?

Key findings

• The paper constructs a comprehensive list of multi-quadratic quad equations satisfying the factorized-discriminant hypothesis, including previously known integrable cases and new models.
• All constructed models, except the primary one (counterpart of Q4), admit Bäcklund transformations connecting them to equations in the multi-affine ABS list.
• The discriminant factorization property enables reformulation of multi-valued quad equations into single-valued systems through auxiliary edge variables and augmented initial data.
• Multidimensional consistency is confirmed for all models, confirming their integrability despite higher-degree nonlinearity.
• The discriminant properties (65) provide a unified algebraic framework for constructing Bäcklund transformations between multi-quadratic and multi-affine models.
• The primary model, the multi-quadratic counterpart of Q4, remains unconnected to the ABS list, suggesting it may represent a genuinely new integrable class.