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Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. (English) Zbl 0846.34006

The authors study Bäcklund transformations and solution hierarchies for the fourth Painlevé equation

$\left({P}_{IV}\right)\phantom{\rule{85.35826pt}{0ex}}w{w}^{\text{'}\text{'}}=\frac{1}{2}{\left({w}^{\text{'}}\right)}^{2}+\frac{3}{2}{w}^{4}+4z{w}^{3}+2\left({z}^{2}-\alpha \right){w}^{2}+\beta ,$

where ${}^{\text{'}}=d/dz$ and $\alpha$ and $\beta$ are arbitrary constants.

They first review Bäcklund transformations for ${P}_{IV}$ given by Lukashevich, Fokas et al., Kitaev and others and they notice that all these transformations are some compositions of those due to Lukashevich which are called $L$-type transformations. However, the transformations due to Kitaev which are called Kitaev fractional transformations are important in obtaining solutions in the hierarchies by only algebraic means.

The authors next give some solution hierarchies: the complementary error function hierarchy, the complex complementary error function hierarchy, the half-integer hierarchy and rational solution hierarchies, where the complementary error function $\text{erfc}\left(z\right)$ is a function defined by

$\text{erfc}\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_{z}^{\infty }exp\left(-{t}^{2}\right)dt·$

Here a solution hierarchy is a set of particular solutions (which may contain an arbitrary constant) of ${P}_{IV}$ for various values of $\alpha$ and $\beta$ under a one-parameter family condition with a set of Bäcklund transformations among the solutions. It should be remarked that the solutions in each hierarchy are derived from a small set of solutions which are called ‘seed solutions’.

Lastly, they introduce the notion of a connected triangle in a solution hierarchy. One can obtain all solutions in the hierarchy from seed solutions in the connected triangle by only algebraic manipulations.

Reviewer: K.Takano (Kobe)
##### MSC:
 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34M55 Painlevé and other special equations; classification, hierarchies 33E30 Functions coming from differential, difference and integral equations