2013/11/30 by S. Ya. Startsev, Sergey Ya. Startsev
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Curvature #Darboux integral #Geometry #Homogeneous space #Injective function #Integrable system #Invertible matrix #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Symmetry (geometry) #Transformation (genetics) #math-ph #math.MP #msc:37K05 #msc:37K10 #msc:37K35 #msc:39A14 #nlin.SI
paper · pdf · doi:10.3842/sigma.2014.066
published as SIGMA 10 (2014), 066, 13 pages
openalex publication_date 2014/06/16 · arxiv created 2014/06/17 · arxiv updated 2014/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This article is devoted to the partial difference quad-graph equations that can be represented in the form (u(i+1, j), u(i+1, j +1)) = (u(i, j), u(i, j +1)), where the map (w, z) ((w, z), (w, z)) is injective. The transformation v(i, j) = (u(i, j), u(i, j + 1)) relates any of such equations to a quad-graph equation. It is proved that this transformation maps Darboux integrable equations of the above form into Darboux integrable equations again and decreases the orders of the transformed integrals by one in the j-direction. As an application of this fact, the Darboux integrable equations possessing integrals of the second order in the j-direction are described under an additional assumption. The transformation also maps symmetries of the original equations into symmetries of the transformed equations (i.e. preserves the integrability in the sense of the symmetry approach) and acts as a difference substitution for symmetries of a special form. The latter fact allows us to derive necessary conditions of Darboux integrability for the equations defined in the first sentence of the abstract.