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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 $\text{P}_{\text{III}}$: $$y''={1\over y} (y')^2- {1\over x} y'+{1\over x} (\alpha y^2+\beta)+\gamma y^3+{\delta\over y}, \Biggl('={d\over dx}\Biggr),\tag1$$ where $\alpha$, $\beta$, $\gamma$ and $\delta$ are arbitrary constants. A survey of the study of Painlevé equations is given in Section 1. The integration of the continuous $\text{P}_{\text{III}}$ with $\beta=\delta=0$ or $\alpha=\gamma=0$ 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 $\text{P}_{\text{III}}$ are described. Section 4 is devoted to the parameter sets for which exact solutions of $\text{P}_{\text{III}}$ 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 $\text{P}_{\text{III}}$ (d-P$_{\text{III}}$) $$y_{n+1}y_n= {\nu y^2_n-\beta x_ny_n-\delta x^2_n\over\gamma x^2_ny^2_n+\alpha x_ny_n+\nu}\tag2$$ is considered. The rational solution of (2) with $\gamma\delta\ne 0$ and exact solutions of (2) with $\gamma$ and $\delta$ 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.

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.)
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