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Bäcklund transformations and solution hierarchies for the third Painlevé equation. (English) Zbl 0878.34006

The authors discuss Bäcklund transformations and solution hierarchies for the third Painlevé equation P III :

y '' =1 y(y ' ) 2 -1 xy ' +1 x(αy 2 +β)+γy 3 +δ y, ' = d dx,(1)

where α, β, γ and δ are arbitrary constants.

A survey of the study of Painlevé equations is given in Section 1. The integration of the continuous P III with β=δ=0 or α=γ=0 and several other properties are reviewed in Section 2, and many scaling transformations are also shown therein. In Section 3, various Bäcklund type transformations for P III are described. Section 4 is devoted to the parameter sets for which exact solutions of P III exist. In Section 5, these exact solutions are categorized into three hierarchies: solutions rational in x; solutions can be expressed by Bessel functions; and solutions rational in x 1/3 .

In Section 5, the following discrete analogy of P III (d-P III )

y n+1 y n =νy n 2 -βx n y n -δx n 2 γx n 2 y n 2 +αx n y n +ν(2)

is considered. The rational solution of (2) with γδ0 and exact solutions of (2) with γ and δ being zero are considered. A final conclusion is stated in Section 6.

This long paper is interesting because the third Painlevé equation has a large number of physically significant applications. A bibliography of 78 papers is included.

MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
39A10Additive difference equations
34A99General theory of ODE
39A12Discrete version of topics in analysis
44A15Special transforms (Legendre, Hilbert, etc.)