A large family of classical special functions are associated with a linear ordinary differential equation with polynomial or rational coefficients. Typical examples are Gauss' hypergeometric functions, Kummer's confluent hypergeometric functions, and various special functions with the name of Airy, Bessel, Hermite, etc. These special functions satisfy a second order linear ordinary differential equations with at most two or three poles on the Riemann sphere.
Extending this family is obviously rather straightforward: Just increase the number and/or order of poles of the coefficients, the order of the equation itself, and even the number of variables. As the general theory of linear ordinary equations in the complex domain was refined by B. Riemann and L. Fuchs, this line of approach to special functions flourished throughout the second half of the 19th century.
Another family of special functions, also discovered in the 19th century, are elliptic functions and their relatives, i.e., theta functions and modular functions click here for the history of elliptic functions). Extending this family of special functions was a central issue of mathematics in the 19th century. Riemann studied this issue, too, and established the theory of Abelian functions in his his geometric framework. .
Painlevé's work may be thought of as a combination of these two lines of approaches to special functions. A clue is the fact that elliptic functions are also characterized by a differential equation. For instance, the Weierstrass p-function is well known to satisfy the first order differential equation
Jacobi's three elliptic functions, too, satisfy a first order system of differential equations. that single out the solution. (These equations are essentially the same as the equations of motion of the Euler-Poinsot rigid body). Unlike the hypergeometric functions etc., however, these differential equations are nonlinear. Just like those elliptic functions, Painlevé aimed to define new special functions as solutions of a suitable nonlinear differential equation.
A question emerges here: What kind of nonlinear differential equations are indeed ``suitable''? A guiding principle can be found in the differential equations that the elliptic functions satisfy. One of the difficulties in analyzing a nonlinear differential equation lies in the fact that the position of singularities of solutions can depend on the solution under consideration (or, equivalently, on the initial values). Such singularities are referred to as ``movable singularities''. Furthermore, solutions of a nonlinear differential equation, in general, can have very bad singularities, such as natural boundaries. If movable singularities are inevitable, the best possible option will be to require that those singularities be as simple as possible. This is indeed the case for the differential equations of elliptic functions: All singularities of elliptic functions are poles, though these poles are ``movable'' in the aforementioned meaning.
Bearing these facts in mind, Panlevé proposed to classify all second order differential equation of the form
Actually, the irreducibility of the Painlevé equations was controversial in the days of Painlevé. R. Liouville argued, without showing any proof, that the first Painlevé was reducible to a forth order linear ordinary differential equation. Painlevé counterattacked and entered the controversy. This issue remained open for many years, and was finally settled in the last decade by Japanese mathematicians (K. Nishioka, H. Umemura, M. Noumi, and Y. Murata).
The concept of Painlevé's work is also very similar to S. Kovalevskaya's classification of ``integrable'' rigid bodies (click here for the history of integrable systems including Kovalevskaya's work). Kovalevskaya's method was to determine the cases where all solutions of the equations of motion have no singularity other than poles. This led to the discovery of the third (and final) example of integrable rigid bodies, which is now called the Kovalevskaya top.
|P. Painlevé||Memoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. Phys. France 28 (1900), 201-261.|
|Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 21 (1902), 1-85.|
|Painlevé reported his results in these papers. The first three of the Painlevé equations were presented here, but the rest were overlooked.|
|B. Gambier||Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critique fixés, Acta. Math. 33 (1910), 1-55.|
|Gambier reexamined Painlevé's previous calculations, and added the remaining three equations. Painlevé's classification program was thus completed.|
Chazy attempted to generalize the work of Painlevé and Gambier to third order differential equations. Chazy's work is closely related to the theory of modular functions. Modular functions are an important family of special functions that satisfy a third order differential quation. Actually they do not have the Painlevé property, because modular functions have movable natural boundaries. Chazy discovered a third order differential equation with the Painlevé, solutions of which are of course not modular, but obeys nearly modular transformations.
|J. Chazy||Sur les équations différentielles dont l'intégrale générale possede un coupure essentielle mobile, C.R. Acad. Sci. (Paris) 150 (1910), 456-458.|
|Sur les équations différentielles de troisième ordre et d'ordre supérieur dont l'ntégrale générale a ses points critiques fixés, Acta Math. 33 (1911), 317-385.|
Chazy's third order equation has a special solution that is also related to the sixth Painlevé equation. This special solution is related to complete period integrals (thereby to modular functions as well). The so called ``Picard solution'' of the sixth Painlevé equation is also realized by complete elliptic integrals and an elliptic function. These special solutions of the sixth Painlevé equations are linked with the following work of R. Fuchs:
|R. Fuchs||Sur quelques équations différentielles linéaires du second ordre, C. R. Acad. Sci. (Paris) 141 (1905), 555-558.|
|Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singuläre Stellen, Math. Ann. 63 (1907), 301-321.|
R. Fuchs (son of L. Fuchs) reported two significant discoveries in this paper.
The first discovery is a new interpretation of the sixth Painlevé equation in the languague of monodromy. He took a second order Fuchsian linear ordinary differential equation on the Riemann sphere, and considered the deformations of this lienar equation that leave invariant the monodromy around regular singular points. Such deformations are nowadays called ``isomonodromic deformations''. Fuchs noticed that those deformations were described by a nonlinear differential equation in suitably selected variables, and pointed out that this nonlinear equation is nothing but the sixth Painlevé equation.
Another discovery is to rewrite the sixth Painlevé equation in terms of an incomplete elliptic integral. In a sense, this is an extension of the Picard-Fuchs equation for complete elliptic integrals. Inspired by this remark, Painlevé proposed yet another expression of his sixth equation using an elliptic function:
|P. Painlevé||Sur les équations différentielles du second ordre à point critiques fixes, C.R. Acad. Sci. (Paris) 143 (1906), 1111-1117.|
These expression of Fuchs and Painlevé revived in Yu. Manin's work more than ninety years later (Yu. I. Manin, Sixth Painlevé equation, universal elliptic curve, and mirror of P2, AMS Transl. (2) 186 (1998), 131-151).
The discovery of isomonodromic deformations indicated a new direction for extending the Painlevé equations. Those extensions were soon attempted by L. Schlesinger and R. Garnier:
|L. Schlesinger||Über eine Klasse von Differentialsystemen beliebliger Ordnumg mit festen kritischer Punkten, J. fÜr Math. 141 (1912), 96-145.|
|R. Garnier||Sur des équations différentielles du troisième ordre dont l'intégrale est uniform et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses point critiques fixés, Ann. Sci. de l'ENS 29 (1912), 1-126.|
|Etudes de l'intégrale générale de l'équation VI de M. Painlevé dans le voisinage de ses singularité transcendentes, Ann. Sci. Ecole Norm. Sup. (3) 34 (1917), 239-353.|
Garnier also noted that an integrable system, solvable by hyoerelliptic Abelian functions, can be obtained by slightly modifying the setup of Schlesinger's isomonodromic deformations:
|R. Garnier||Sur une classe de systèmes differentiels abéliens deduits de la théorie des équations linéaires, Rend. Circ. Mat. Palermo 43 (1918-19), 155-191.|
|This paper of Garnier is one of the earliest papers that pointed out a hidden relation between integrable systems and equations of Painlevé type (click here for the history of integrable systems).|
The point of view of isomonodromic deformations experienced a dramatic revival due to M. Jimbo, T. Miwa, Y. Môri and M. Sato in the late seventies (M. Sato, T. Miwa and M. Jimbo, Holonomic quantum fields II, Publ. RIMS, Kyoto Univ., 15 (1979), 201-27; M. Jimbo, T. Miwa, Y. Môri and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica 1D (1980), 80-158) soon after T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch encountered the Painlevé equation in the Ising model of statistical physics (E. Barouch, B.M. McCoy and T.T. Wu, Zero-field susceptibility of the two-dimensional Ising model near Tc, Phys. Rev. Lett. 31 (1973), 1409-1411; T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin-spin correlation functions for the two-dimensional Isong model, Phys. Rev. B13 (1976), 316-374).
Painlevé himself pursued his equations in a different directions, in particular, in the language of algebraic geometry of surfaces. The famous Stockholm lectures was intended to present such ideas.
|P. Painlevé||Leçons de Stockholm, Oeuvre t.I (C.N.R.S., Paris, 1972).|
The algebro-geometric point of view is also a keynote of recent researches on the Painlevé equations, which originate in K. Okamoto's work in the late seventies (K. Okamoto, Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. 5 (1979), 1-79).