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The Painlevé connection problem: An asymptotic approach. I. (English) Zbl 0761.34003

The author clearly and completely develops a rich mathematical structure surrounding the Painlevé equations which is closely related to the well known developed direct methods for solving nonlinear partial differential equations by the implementation of linear integral equations. The integral equation has a nonsingular kernel and the solutions are obtained via a linear integral equation of the form

$𝒦\left(x,y;t\right)=ℱ\left(x+y;t\right)+{\int }_{x}^{\infty }𝒦\left(x,z;t\right)𝒩\left(x;z,y;t\right)dx,$

where $ℱ$ satisfies appropriate conditions and then the function $\mu =\frac{d}{dx}𝒦\left(x,x;t\right)$ satisfies the original nonlinear PDE. This paper has the similar back drop in place but adds to this remarkable result. It studies in detail a solution to the Painlevé connection problem by investigating the asymptotic behavior of a solution near a fixed singular point to its behavior near some other (or the same) such point. This additional connection can be solved directly for the first two Painlevé equations through a natural asymptotic method. The construction and significant details are all included in the paper. In particular, a uniformly valid description of the general asymptotic behavior given to leading order by elliptic functions is carefully derived by a generalization of the so called multiple-scales method. The paper carefully develops the results for the first two Painlevé equations ${P}_{I}$ and ${P}_{II}$ having the form $f{g}^{\text{'}}{}^{2}{u}_{zz}+\left(2{f}^{\text{'}}{g}^{\text{'}}+f{g}^{\text{'}\text{'}}\right){u}_{z}+{f}^{\text{'}\text{'}}u=\frac{3}{2}\left({f}^{2}{u}^{2}-x\right)$ and $f{g}^{\text{'}}{}^{2}{u}_{zz}+\left(2{f}^{\text{'}}{g}^{\text{'}}+f{g}^{\text{'}\text{'}}\right){u}_{z}+{f}^{\text{'}\text{'}}u=2\left({f}^{3}{u}^{2}-xfu\right)+{a}_{0}$. Asymptotic behaviors are then one of the main focuses of the paper. It concludes with investigating ${P}_{1}$ near the imaginary $z$-axis and differential equations satisfied by complete elliptic integrals.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies