In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by Émile Picard (1889), Paul Painlevé (1900, 1902), Richard Fuchs (1905), and Bertrand Gambier (1910).
History
Origins Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second-order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is rare in nonlinear equations. Poincaré and Lazarus Fuchs showed that any first order equation (that is, an ODE involving only up to the first derivative) with the Painlevé property can be transformed into the Weierstrass elliptic equation or the Riccati equation, all of which can be solved explicitly in terms of integration and previously known special functions. Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found in Picard (1889) a special case of what was later called Painleve VI equation (see below). (For orders greater than 2 the solutions can have moving natural boundaries.) Specifically, let φ {\textstyle \varphi } be the elliptic function defined by φ : y ↦ φ ( y , x ) , y = ∫ ∞ φ d z z ( z − 1 ) ( z − x ) {\displaystyle \varphi :y\mapsto \varphi (y,x),\qquad y=\int _{\infty }^{\varphi }{\frac {\mathrm {d} z}{\sqrt {z(z-1)(z-x)}}}}
and let ω 1 ( x ) , ω 2 ( x ) {\textstyle \omega _{1}(x),\omega _{2}(x)} be its two half-periods. Then the function u : x ↦ u ( x ) = φ ( 2 c 1 ω 1 ( x ) + 2 c 2 ω 2 ( x ) , x ) {\displaystyle u:x\mapsto u(x)=\varphi \left(2c_{1}\omega _{1}(x)+2c_{2}\omega _{2}(x),x\right)} with ( c 1 , c 2 ) {\textstyle \left(c_{1},c_{2}\right)} arbitrary constants satisfies the Painleve VI equation in the case of α = β = γ = δ − 1 / 2 = 0 {\textstyle \alpha =\beta =\gamma =\delta -1/2=0} .
Classification Around 1900, Paul Painlevé studied second-order differential equations with no movable singularities. He found that up to certain transformations, every such equation of the form
y ′ ′ = R ( y ′ , y , t ) {\displaystyle y^{\prime \prime }=R(y^{\prime },y,t)}
(with R {\displaystyle R} a rational function) can be put into one of 50 canonical forms (listed in (Ince 1956)). Painlevé (1900, 1902) found that 44 of the 50 equations are reducible, in the sense that they can be solved in terms of previously known functions, leaving just 6 equations requiring the introduction of new special functions to solve them. These six second order nonlinear differential equations are called the Painlevé equations and their solutions are called the Painlevé transcendents. There were some computational errors, and as a result he missed 3 of the equations, including the general form of Painleve VI. Painlevé's student Bertrand Gambier fixed the errors and completed the classification. Independently of Painlevé and Gambier, equation Painleve VI was found by Richard Fuchs from completely different considerations: he studied isomonodromic deformations of linear differential equations with regular singularities. The most general form of the sixth equation was missed by Painlevé, but was discovered in 1905 by Richard Fuchs (son of Lazarus Fuchs), as the differential equation satisfied by the singularity of a second order Fuchsian equation with 4 regular singular points on the projective line P 1 {\displaystyle \mathbf {P} ^{1}} under monodromy-preserving deformations. It was added to Painlevé's list by Gambier (1910).
… excerpt ends here. Continue reading the full article.
