Discrete Nonlinear Equations

Abstract

We first explain why the logistic map, which admits a continuum limit to the Riccati equation, is a bad discretization of the latter. We then present the difficulty to give an undisputed definition for the discrete Painlevé property, and introduce the three main methods of the discrete Painlevé test: the criterium of polynomial growth (Hietarinta and Viallet (Phys Rev Lett 81:325–328, 1998), Halburd (Proc Roy Soc A 473:20160831 (13 pp), 2017)), the singularity confinement method (Grammaticos et al. (Phys Rev Lett 67:1825–1828, 1991)), and the perturbation of the continuum limit (Conte and Musette (Phys Lett A 223:439–448, 1996. arXiv:Solv-int/9610007)). Later, we recall the remark by Baxter and Potts that the addition formula of the Weierstrass function ℘ is an exact discretization of the Weierstrass equation. Finally, we introduce the two main methods able to build discrete Painlevé equations: (1) an analytic method which starts from the addition formula of the elliptic function, takes inspiration from the method of Painlevé and Gambier and produces a rather long, but incomplete, list of discrete Pn equations; (2) a geometric method based on the theory of rational surfaces, which builds ex abrupto the master discrete Painlevé equation ell − PVI, whose coefficients have an elliptic dependence on the independent variable. The main properties of all these d − Pn are summarized. This chapter also includes discrete Ermakov-Pinney equations and discrete nonlinear Schrödinger equations.