an:00051284
Zbl 0752.34003
Gromak, V. I.; Lukashevich, N. A.
Analytic properties of solutions of Painlev?? equations
RU
Minsk: Izdatel'stvo Universitetskoe. 160 p. (1990).
1990
b
34M55 34-02 34A05 30-02 33-02
Painlev?? equations; immobile singular points; explicit representations of meromorphic solutions; Airy, Bessel, Weber-Hermite and hypergeometric functions; nonlinear equations of mathematical physics
Painlev?? equations have the form \(w''(z)=R(w,w')\) where \(R\) is a rational function of \(w'\) and \(w\) with analytic coefficients. It is known that there are six types of Painlev?? equations such that the remaining 44 have solutions expressible via either elementary functions, or solutions of some linear equations, or solutions of first order equations, or, at last, solutions of the afore-said six equations. The book under review contains exactly six chapters, one for each of the distinguished equations. The authors study immobile singular points and solutions in their neighborhoods, give explicit representations of meromorphic solutions, if any, and study relations between solutions of different equations. Some classes of solutions are expressed via Airy, Bessel, Weber-Hermite and hypergeometric functions. Applications are given to nonlinear equations of mathematical physics.
V.A.Tkachenko (Khar'kov)