Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation

Abstract

Blending Painlev\'e property with singularity confinement for a general arbitrary order Sawada-Kotera differential-difference equation, we find a proliferation of ``tau-functions'' (coming from strictly confined patterns). However only one of these function enters into the Hirota bilinear form (the others give multi-linear expressions) but has specific relations with all others. We also discuss the case of two modifications of Sawada-Kotera showing that periodic patterns appear in addition to strictly confined ones. Fully discretisations and express method for computing algebraic entropy are discussed.