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Zbl 0846.34006
Bassom, Andrew P.; Clarkson, Peter A.; Hicks, Andrew C.
Bäcklund transformations and solution hierarchies for the fourth Painlevé equation.
(English)
[J] Stud. Appl. Math. 95, No.1, 1-71 (1995). ISSN 0022-2526; ISSN 1467-9590/e

The authors study Bäcklund transformations and solution hierarchies for the fourth Painlevé equation $$(P_{IV}) \hskip3cm ww''= {\textstyle {1\over 2}} (w')^2+ {\textstyle {3\over 2}} w^4+ 4zw^3+ 2(z^2- \alpha)w^2+ \beta,$$ where ${}'= d/dz$ and $\alpha$ and $\beta$ are arbitrary constants. \par 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. \par 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 $\text {erfc} (z)$ is a function defined by $$\text {erfc} (z)= {2\over {\sqrt {\pi}}} \int^\infty_z \exp (-t^2) dt.$$ 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'. \par 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.
[K.Takano (Kobe)]
MSC 2000:
*34A25 Analytical theory of ODE
34M55 Painlevé and other special equations
33E30 Functions coming from diff., difference and integral equations

Keywords: Bäcklund transformations; solution hierarchies; fourth Painlevé equation; Kitaev fractional transformations; connected triangle

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