Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 084, 7 pages      arXiv:2304.06956      https://doi.org/10.3842/SIGMA.2023.084
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane

Decio Levi a and Miguel A. Rodríguez b
a) Mathematical and Physical Department, Roma Tre University, Via della Vasca Navale, 84, I00146 Roma, Italy
b) Departamento de Física Teórica, Universidad Complutense de Madrid, Pza. de las Ciencias, 1, 28040 Madrid, Spain

Received April 17, 2023, in final form October 23, 2023; Published online November 01, 2023

Abstract
Determining if an $(1+1)$-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to Yamilov's theorem. We shall apply this result to a class of differential-difference equations obtained as partial continuous limits of 3-points difference equations in the plane and conclude that they cannot be integrable.

Key words: difference equations; integrability; Yamilov's theorem.

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