Group Analysis of Differential Equations and Integrable Systems − 2018

Pavlos Xenitidis (School of Mathematics, Statistics & Actuarial Science, University of Kent, UK)

Deautonomization of integrable difference equations

Integrability conditions for difference equations can be used not only to check the integrability of a given equation but also to compute its generalised symmetries and derive conservation laws. In this talk we explore one more application of these conditions: the construction of integrable non-autonomous equations. We demonstrate how the integrability conditions can be used to deautonomize a given autonomous equation and apply this procedure to a certain family of equations defined on an elementary quadrilateral of the ${\mathbb{Z}}^2$ lattice. In particular we show that all the non-autonomous equations we find are related via Miura transformations to the same integrable two-quad autonomous equation. Finally we prove the integrability of the latter equation by generating an infinite hierarchy of its symmetries using a local master symmetry.