nLab Painleve transcendent

Overview of Painlevé transcendents

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

$$u'' = F (z; u, u'),$$

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.

Painlevé VI is 2nd order nonlinear ODE governing the isomonodromic deformations of Fuchsian differential equations

$$\frac{d f}{d z} = \left(\frac{A_1}{z} + \frac{A_2}{z-t}+ \frac{A_3}{z-1}\right),$$

where $A_i\in sl_2(\mathbf{C})$ and parameter $t$ varies through $\mathbf{P}^1\backslash\{0,1,\infty\}$.

Literature and links

General

• eom/Painlevé equation
• wikipedia: Painlevé transcendents, isomonodromic deformation
• One hundred years of PVI, the Fuchs–Painlevé equation, J. Phys. A, special issue, Preface, pdf

Related $n$Lab entries include Picard-Vessiot theory, Riemann-Hilbert problem, integrable system, monodromy, holonomic quantum field

Classical works

• 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

Collection one century later

• The Painlevé property, one century later, Robert Conte, ed. CRM Series in Mathematical Physics, Springer, 1999 doi

Among the papers in the volume:

• Boris Dubrovin, Painlevé transcendents in two-dimensional topological field theory, 287-412

Another collection

• Théories asymptotiques et équations de Painlevé, Séminaires et Congrès 14, Société Mathématique de France 2006

Contemporary works

Modern viewpoint to R. Fuchs work

• Yuri Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf{P}^2$, In: Geometry of Differential Equations, ed. by A. Khovanskii, A. Varchenko, V. Vassiliev. Amer. Math. Soc. Transl. (2), vol. 186 (1998) 131–151 alg–geom/9605010

Connection to mirror symmetry is in

• Yu. I. Manin, Rational curves, elliptic curves, and the Painlevé equation, Surveys in modern mathematics, London Math. Soc. Lec. Note Ser. 321, Cambridge Univ. Press 2005, 24-33 doi MR2166923

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: Painlevé 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, n: Geometry and physics: A Festschrift in honour of Nigel Hitchin, Oxford Univ. Press 2018 arxiv/1705.01447; Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras, Adv. Math. 376 (2021) 107442 arxiv/1905.02772 doi

• Marta Mazzocco, Vladimir Rubtsov, Confluence on the Painlevé monodromy manifolds, their Poisson structure and quantisation, arXiv/1212.6723

• 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

• M. A. Bershtein, A. I. Shchechkin, Bilinear equations on Painlevé $\tau$ functions from CFT, Commun. Math. Phys. 339, 1021–1061 (2015) doi

Multidimensional generalizations of Painlevé VI are introduced in

• G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky, A. Zotov, Painlevé field theory, arXiv/1306.3265
• Nalini Joshi, Pieter Roffelsen, On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem, arXiv/2202.10597
• Nikita Nekrasov, Blowups in BPS/CFT correspondence, and Painlevé VI, arXiv:2007.03646

Model-theoretic aspects

• Joel Nagloo, Geometric triviality of the strongly minimal second Painlevé equations Annals of Pure and Applied Logic, 166(3):358–368, 2015.
• J. Nagloo, A. Pillay, On the algebraic independence of generic Painlevé transcendents, Compos.Math. 150 (2014) 668-678.
• James Freitag, Algebraic relations between solutions of Painlevé equations, arXiv:1710.03304
• Guy Casale, James Freitag, Joel Nagloo, Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups, Annals of Mathematics 192 (2020), 721–765 doi