{"title":"From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations","authors":"Dan-da Zhang, Da-jun Zhang, Peter H. van der Kamp","doi":"10.1007/s11040-021-09406-1","DOIUrl":null,"url":null,"abstract":"<div><p>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. <b>41</b>(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. <b>18</b>, 337–365, 2011), and to a multi-quadratic ABS* equation giving rise to a multi-quartic equation.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Physics, Analysis and Geometry","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s11040-021-09406-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 4
Abstract
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.
期刊介绍：
MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas.
The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process.
The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed.
The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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