Painlevé-Gambier classification

The Painlevé-Gambier classification, often referred to simply as the Painlevé classification, is a categorization of certain types of nonlinear ordinary differential equations (ODEs). This classification, developed by Paul Painlevé and Benoît Gambier in the late 19th and early 20th centuries, focuses on identifying and studying second-order ODEs that have the property of not possessing movable critical points (i.e., singularities whose location depends on the initial conditions of the solutions).

The primary goal of the Painlevé classification is to find all second-order nonlinear ODEs of the form

where $F$ is a rational function of its arguments, that satisfy the Painlevé property: their solutions have no movable singularities other than poles. This property is significant because it implies that the solutions are well-behaved in a certain sense and can be used in various physical and mathematical applications.

The classification identifies six special equations, known as the Painlevé equations ${\text{P}}_{\text{I}}$ through ${\text{P}}_{\text{VI}}$, which are:

1. Painlevé I (P${}_{\text{I}}$):$$y^{″}=6y^2+x$$
2. Painlevé II (P${}_{\text{II}}$):$$y^{″}=2y^3+xy+\alpha$$
3. Painlevé III (P${}_{\text{III}}$):$$y^{″}=\frac{y^{\prime 2}}{y}-\frac{y^{\prime }}{x}+\frac{\alpha y^2+\beta }{x}+\gamma y^3+\frac{\delta }{y}$$
4. Painlevé IV (P${}_{\text{IV}}$):$$y^{″}=\frac{y^{\prime 2}}{2y}+\frac{3}{2}{y}^3+4x{y}^2+2(x^2-\alpha )y+\frac{\beta }{y}$$
5. Painlevé V (P${}_{\text{V}}$):$$y^{″}=(\frac{1}{2y}+\frac{1}{y-1})y^{\prime 2}-\frac{y^{\prime }}{x}+\frac{(y-1{)}^2}{x^2}(\alpha y+\frac{\beta }{y})+\gamma y+\frac{\delta y(y+1)}{y-1}$$
6. Painlevé VI (P${}_{\text{VI}}$):$$y^{″}=\frac{1}{2}(\frac{1}{y}+\frac{1}{y-1}+\frac{1}{y-x})y^{\prime 2}-(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y-x})y^{\prime }+\frac{y(y-1)(y-x)}{x^2(x-1{)}^2}(\alpha +\beta \frac{x}{y^2}+\gamma \frac{x-1}{(y-1{)}^2}+\delta \frac{x(x-1)}{(y-x{)}^2})$$

These equations are notable for their complex behavior and their connections to various areas of mathematics and physics, including integrable systems, random matrix theory, and statistical mechanics.

The Painlevé-Gambier classification extends beyond these six Painlevé equations to include other related equations that share similar properties, but the six equations listed above are the most famous and widely studied.