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 (this is more effective way to express that they can be made single-valued functions by uniformization). 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 (automatically, $F$ is at most quadratic in $u,u'$) 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.
Related $n$Lab entries include Picard-Vessiot theory, Riemann-Hilbert problem, integrable system, monodromy, holonomic quantum field
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 141 (1905) 555–558
B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critiques fixes Acta Math. 33 (1910) 1–55
Among the papers in the volume:
Modern viewpoint to R. Fuchs work
Connection to mirror symmetry is in
An arithmetic analogue is studied in
Alexandru Buium, Yuri I. Manin, Arithmetic differential equations of Painlevé VI type, in: Arithmetic and Geometry, Luis Dieulefait, Gerd Faltings, D. R. Heath-Brown, B. Z. Moroz, Cambridge Univ. Press 2015, arxiv/1307.3841
Emma Previato, Curves in isomonodromy and isospectral deformations: Painlev'e VI as a case study, In: Algebraic curves and their applications. Contemporary Math. 724 (2018) 258-276, Amer. Math. Soc.
M. Olshanetsky, Painlevé type equations and Hitchin systems, Proc. “Integrability: the Seiberg-Witten and Whitham Equations”, Edinburgh, Sep. 1998 math-ph/9901019
AM Levin, MA Olshanetsky, Hierarchies of isomonodromic deformations and Hitchin systems, 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
Hiroshi Umemura, On the irreducibility of the first differential equation of Painlevé, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo (1988) 771–789; Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J. 117, 125-171 doi
Robert Conte, Micheline Musette, The Painlevé handbook
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
M. V. Babich, Isomonodromic deformations and Painlevé equations, Constr. Approx. 41:3 (2015) 335-356 doi
Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, IMRN 24 (2017) 7639–7691 arxiv/1511.03851 doi; Algebras of quantum monodromy data and decorated character varieties, arxiv/1705.01447; Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras, arxiv/1905.02772
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
Kazuo Okamoto, Polynomial Hamiltonians associated with Painlevé equations, I, Proc. Jap. Acad. Ser. A, Math. Sci. 56:6 doi; II, Differential equations satisfied by polynomial Hamiltonians, Proc. Jap. Acad. Ser. A, Math. Sci. 56:8 doi euclid
H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys. 220 (2001) 165–229
Multidimensional generalizations of Painlevé VI are introduced in
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