What is this thing called ``Painlevé''?

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ODEs and Singularities

Consider the ordinary differential equation (ODE)

y'=-y^2

The solution is simple to find by separation of variables. Divide both sides by y^2 and integrate both sides. This leads to the solution

y=1/(x-a)

where a is a free constant. This is a ``general'' solution (because it has one free constant, corresponding to the order of the equation). There is also a ``special'' solution:

y=0.

The solutions we have found are analytic except for the general solution at the point x=a. This point is movable! Its location changes with the initial condition that defines a particular solution. E.g.,

y(0)=1    => a=-1,
y(0)=-2 => a=1/2.

And, this movable singularity is a pole.

Excercise: Solve the ODE y'=-y^3 and show that the general solution is multivalued around its movable singularities.

Now, suppose we were given a different ODE

y'=- y^2/x

The general solution is now

y=1/(log(x)-a)

This solution has a logarithmic singularity at x=0 as well as a pole at x=exp(a). Notice that the first singularity, at x=0, does not change with initial conditions. It is fixed by the fact that x=0 occurs as a singularity in the equation itself.

The property that all movable singularities of all solutions are poles is called the Painlevé property. Both ODEs above have this property.

Back to Contents Nalini Joshi
Last modified: 29 November 2001 by
N.Joshi