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From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations


We provide a method which from a given auto-Bäcklund transformation (auto-BT) produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions of ABS equations and explains the origin of each auto-BT listed in Atkinson (J. Phys. A: Math. Theor. 41(8pp), 135202, 2008). The method is also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes which have not been found in Boll (J. Nonl. Math. Phys. 18, 337–365, 2011), and to a multi-quadratic ABS* equation giving rise to a multi-quartic equation.

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The authors are thankful for detailed comments and questions from the referees. Financial support was provided by a La Trobe University China studies seed-funding research grant, by the department of Mathematics and Statistics of La Trobe University, by the NSF of China [Grants 11631007, 11875040, 11801289], by the Science and technology innovation plan of Shanghai [No.20590742900], and by the K.C. Wong Magna Fund in Ningbo University.

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Correspondence to Peter H. van der Kamp.

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Communicated by:F W Nijhoff

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Zhang, Dd., Zhang, Dj. & van der Kamp, P. From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations. Math Phys Anal Geom 24, 33 (2021).

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  • 3D consistency
  • Auto-Bäcklund transformation
  • Quad equation
  • Superposition principle
  • Tetrahedron property
  • Planar
  • Torqued equation

Mathematics Subject Classification (2020)

  • 37K60