Stokes' phenomenon in continuous limits of discrete Painlevé I

Abstract: We use exponential asymptotic analysis to identify the relevance of Stokes' phenomenon to integrability in discrete systems. We study Stokes' phenomenon in two discrete problems with the same (leading-order) continuous limit, a finite-difference discretisation of the first continuous Painlev\'{e} equation and the first discrete Painlev\'{e} equation, as well as a family of differential equation associated with each discrete problem. This analysis reveals two important observations. Firstly, the orderly behaviour that characterises Stokes' phenomenon in discrete equations emerges naturally from corresponding continuous differential equations as the order of the latter increases, although this is not apparent at low orders. Secondly, Stokes' phenomenon vanishes in the continuum limit of the integrable discrete equation, but not the non-integrable discrete equation. This means that subdominant exponentials do not appear in the integrable equation, and therefore do not cause moveable singularities to form in the solution. The results are clarified further by consideration of one-parameter family of difference equations that interpolates between the two considered in detail.