A1 Refereed original research article in a scientific journal
Search for CAC-integrable homogeneous quadratic triplets of quad equations and their classification by BT and Lax
Authors: Jarmo Hietarinta
Publisher: TAYLOR & FRANCIS LTD
Publication year: 2019
Journal: Journal of Nonlinear Mathematical Physics
Journal name in source: JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS
Journal acronym: J NONLINEAR MATH PHY
Volume: 26
Issue: 3
First page : 358
Last page: 389
Number of pages: 32
ISSN: 1402-9251
DOI: https://doi.org/10.1080/14029251.2019.1613047
Abstract
We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of "multidimensional consistency" (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using "Consistency-Around-a-Cube" (CAC). In this context it is often assumed that the equations on the six sides of the cube are the same (up to lattice parameters), but this assumption was relaxed in the classification of Boll [3]. We present here the results of a search and classification of homogeneous quadratic triplets of multidimensionally consistent lattice equations, allowing different equations on the three orthogonal planes (hence triplets) but using the same equation on parallel planes. No assumptions are made about symmetry or tetrahedron property. The results are then grouped by subset/limit properties, and analyzed by the effectiveness of their Backlund transformations, or equivalently, by the quality of their Lax pair (fake or not).
We consider two-dimensional lattice equations defined on an elementary square of the Cartesian lattice and depending on the variables at the corners of the quadrilateral. For such equations the property often associated with integrability is that of "multidimensional consistency" (MDC): it should be possible to extend the equation from two to higher dimensions so that the embedded two-dimensional lattice equations are compatible. Usually compatibility is checked using "Consistency-Around-a-Cube" (CAC). In this context it is often assumed that the equations on the six sides of the cube are the same (up to lattice parameters), but this assumption was relaxed in the classification of Boll [3]. We present here the results of a search and classification of homogeneous quadratic triplets of multidimensionally consistent lattice equations, allowing different equations on the three orthogonal planes (hence triplets) but using the same equation on parallel planes. No assumptions are made about symmetry or tetrahedron property. The results are then grouped by subset/limit properties, and analyzed by the effectiveness of their Backlund transformations, or equivalently, by the quality of their Lax pair (fake or not).
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