On a transformation of the Painlevé III and V equations. (Russian. English summary) Zbl 0845.34009

Lukashevich, N. A.; Chichurin, A. V.

Izv. Akad. Nauk Belarusi, Ser. Fiz.-Mat. Nauk 1995, No. 2, 9-12 (1995).

Summary: We apply an analytic method to III and V Painlevé equations. This method reduces the Painlevé equations to an equation of the form \[ {d^2 w\over dv^2}= K(w, v)^{- 1}\Biggl( E(w, v)\Biggl({dw\over dv}\Biggr)^3+ F(w, v)\Biggl({dw\over dv}\Biggr)^2+ G(w, v) {dw\over dv}+ H(w, v)\Biggr), \] where \(E\), \(F\), \(G\), \(H\), \(K\) are polynomials of \(w\), \(v\); \(w= w(z)\), \(v= v(z)\) are analytic functions of \(z\) at the field \(\mathfrak D\).

MSC:

34M55

Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies

Keywords:

analytic method; III and V Painlevé equations