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 generally propagate with change of initial conditions. Very important is if the singularities do not move or monodromies don’t change with change of parameters. An ordinary differential equation (ODE) satisfies the Painlevé property if all solutions are single valued around every movable singularity. 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 second order ODEs of the form
where $F$ is rational in $u,u'$ and analytic in $z$ and which satisfy the Painlevé property 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 (44+6) up to a number of standard transformations. Painlevé transcendents are now of central importance in the study of integrable systems.
There are also some noncommutative versions which are still purely understood.
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.
Richard Fuchs, Comptes Rendus de l’Académie des Sciences Paris 1905 141, 555–558
wikipedia: Painlevé transcendents
One hundred years of PVI, the Fuchs–Painlevé equation, J. Phys. A, special issue, Preface, pdf
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
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
A. Okounkov, E. Rains, Noncommutative geometry and Painlevé equations, Algebra & Number Theory 9(6) 1363–1400 (2015) doi
Multidimensional generalizations of Painlevé VI are introduced in
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