In the study of ordinary differential equations one of the important things is the behaviour of monodromies and closely related singularities of solutions. In the linear case, the poles possibly appear just at the poles of coefficients of the solutions. In the nonlinear case the solutions can appear elsewhere and propagate with change of parameters involved in coefficients. Very important is if the singularities do not move and or monodromies don’t change with change of parameters. A class of such “good” nonlinear equations has been defined by Paul Painlevé (wikipedia) who discovered at the end of 19th century a truly remarkable fact that all such equations have solutions which can be expressed in terms of well known functions like elementary and hypergeometric functions and only 6 new kinds of transcendental functions called Painlevé I-VI. Furthermore he obtained a complete classification of such equations (of second order?) in 50 classes up to a number of standard transformations. Painlevé transcendents are now of central importance in the study of integrable systems.

There are also noncommutative versions.

Literature and links

P. Painlevé, Sur les équations differentielles du second ordre et d’ordre superieur, dont l’integrale génerale est uniforme, Acta Math. 25 (1902), pp. 1–86.

A. A. Kapaev, Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37, 11149 (2004) doi

A. A. Bolibruch, A. R. Its, A. A. Kapaev, On the Riemann–Hilbert–Birkhoff inverse monodromy problem and the Painlevé equations, Алгебра и анализ, 16:1 (2004), 121–162

Marco Bertola, Fredholm determinants and (noncommutative) Painlevé II equation, slides, pdf

Revised on November 28, 2011 20:30:40
by Zoran Škoda
(161.53.130.104)