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Rational solutions of the fifth Painlevé equation. (English) Zbl 0804.34006

Painlevé and his school classified all differential equations of the form ${y}^{\text{'}\text{'}}=F\left(z,y,{y}^{\text{'}}\right)$ on the complex plane. They found 50 equations in this milieu where $F$ is rational in $y$ and ${y}^{\text{'}}$, locally analytic in $z$, and for each solution all the singularities which are dependent on the initial conditions are poles. 44 of these equations can be integrated in terms of classical functions and transcendents, elliptic functions or transformed into the remaining six equations now called the Painlevé equations. The author gives a remarkable analysis of the fifth equation

${y}^{\text{'}\text{'}}=\left(\frac{1}{2y}+\frac{1}{y-1}\right){\left({y}^{\text{'}}\right)}^{2}-\frac{{y}^{\text{'}}}{z}+\frac{{\left(y-1\right)}^{2}}{{z}^{2}}\left(\alpha y+\frac{\beta }{y}\right)+\frac{\gamma y}{z}+\frac{\delta y\left(y+1\right)}{y-1}·$

The paper obtains necessary and sufficient conditions for the existence of rational solutions for this equation. The Murata approach on the second and fourth Painlevé equation is the technique implemented in this paper. Each category of the parameters are clearly identified and then several theorems are clearly proven within the category. Each proof is concise and contains enough information so that the reader can clearly understand the particular category. As an illustration of the types of theorems proven, we cite the following: the function $h\left(z\right)=h\left(z,\stackrel{\to }{v}\right)$ satisfies the equation ${\left(z{h}^{\text{'}\text{'}}\right)}^{2}-{\left[h-z{h}^{\text{'}}+2{\left({h}^{\text{'}}\right)}^{2}\right]}^{2}+4{\prod }_{k=1}^{4}\left({h}^{\text{'}}-{v}_{k}\right)=0$ if and only if a solution $\left({y}_{1},{y}_{2}\right)$ of the Painlevé System $H={H}_{5}\left(\stackrel{\to }{v}\right)$ is as follows: ${y}_{1}=\frac{1}{2{A}_{3}{A}_{4}}\left\{z{h}^{\text{'}\text{'}}+h-z{h}^{\text{'}}+2{\left({h}^{\text{'}}\right)}^{2}\right\}$, ${y}_{2}=\frac{1}{2{A}_{2}}\left\{z{h}^{\text{'}\text{'}}-h+z{h}^{\text{'}}-2{\left({h}^{\text{'}}\right)}^{2}\right\}$, where ${A}_{k}={h}^{\text{'}}+{v}_{k}$ $\left(k=2,3,4\right)$.

MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34A34 Nonlinear ODE and systems, general