nLab movable singularity

Given an ordinary differential equation (ODE) for an analytic function from a domain in the complex plane (or on a Riemann surface) to the complex numbers, a point is a fixed singularity if it is a singular point of every solution of the ODE and a movable singularity if the position of the singular point depends on the solution (or equivalently on the initial conditions).

For example, the Ricatti equation $u'' + u^2 = 0$ has a general solution $u(z) = 1/(z-z_0)$, where $z_0\in\mathbb{C}$ is an arbitrary constant. The position of the pole is at $z_0$, hence it depends on a particular solution.

An ODE satisfies the Painlevé property if all solutions are single-valued around every movable singularity.

Related entries

Related entries include monodromy, holomorphic function, Riemann-Hilbert problem, Painlevé transcendent, Milnor fiber, Stokes phenomenon, semiclassical approximation, Fuchsian equation, hypergeometric function, and Picard-Lefschetz theory.

References

• Martin D. Kruskal, Nalini Joshi, Rod Halburd, Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property, 1996 pdf

• Henryk Żołądek, The monodromy group, Monografie Matematyczne 67, 588 pp. Birkhäuser 2006

• Vladimir V. Golubev, Lekcii po analitičeskoi teorii differencial’nyh uravnenii. 1ed 1941, 2nd ed. Moscow-Leningrad. 1950; German transl. Vorlesungen ueber Differentialgleichungen im Komplexen, Deutsche Verlag Wissenschaft., 1958

As a possible generalization for PDEs, one can consider analytic movable singularity manifolds as in

• J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, J. Math. Phys. 24, 522–526 (1983)

category: analysis