Painlevé equations are used to solve non-linear second order differential equation. For more on Painleve’s work see this and for short refresher on Painlevé equations, Encylopedia of Mathmematics.

This note concerns more on the Discrete derivatives of Painlevé’s Equations.

Integratibility is not a well defined term. Poincaré’s definition of Integrabiliy is – to integrate a DE is to find for the general solution, a finite expression, in a finite number of funtions. i.e Solution should be a single value.

Soliton Systems for e.g are single valued :

Multi valued functions of a complex variable have branch points, i.e critical points. While implies critical singularities of a linear ODE are fixed

We can derive new functions from Non-linear equations :

solution:
Singular at is said to be fixed.

e.g

1. 
   solution:
2. 
   solution:

Singularities of a nonlinear systems generally do not show Painlevé Property*

*Painlevé Property
An ODE is said to possess the Painlevé property if all its solution are single valued. Beauty of having this property is it allows the transformation of Painlevé Equations into Riccati Equations, which could be solved using known techniques.
Riccati Equations are fo the form :

— (I)

Painlevé Classified Equatios into 6 types (Transcedants). They were mostly forgotten but their revival was necessitated after soliton systems.

 

Dicrete Painlevé was discovered mostly due to efforts on Quantum Gravity. See Discrete Painlevé equations and their appearance in quantum gravity by AS Fokas et al.

On discretisation, Riccati Equation given above at  (I) becomes

A caveat though, Non-linear discretisation is not unique and may result in incorrect functions. Discretisation should be fo rational form and have homographic mapping. There are more than 20 discrete Painlevé Equations.

Solutions to Discrete Painlevé Equations can be found on this paper by A.Ramani et al