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Zbl 0777.34007
Kapaev, A.A.; Kitaev, A.V.
Connection formulae for the first Painlevé transcendent in the complex domain.
(English)
[J] Lett. Math. Phys. 27, No.4, 243-252 (1993). ISSN 0377-9017; ISSN 1573-0530/e

The authors study the first Painlevé equation (1) $y''=6y\sp 2+x$ and the auxiliary linear ordinary differential equation $${d \psi \over d\lambda}= \left\{(4\lambda\sp 2+x+2y\sp 2) \sigma\sb 3-i(4y \lambda\sp 2+x+2y\sp 2)\sigma\sb 2- \left(2y'\lambda+{1 \over 2\lambda} \right) \sigma\sb 1\right\} \psi. \tag 2$$ They then go on using the isomonodrome deformation method to find a complete asymptotic descriptions of the first Painlevé transcendent in the complex domain by relating equations (1) and (2). The starting point is a result due to M. Jimbo and T. Mirva, whereby an analytic function $y(x)$ is a solution of equation (1) if and only if the monodromy data of equation (2) do not depend on $x$. The authors then explore this relationship using the Stokes multipliers $s\sb k$. A very elegant treatment is then presented whereby asymptotic descriptions are presented for the solutions of the equation (1) connecting to the Stokes multiplier constraints in sectors of the complex plane. Several theorems are proven such as if $y(x)$ is a solution of (1) subject to Stokes multiplier constraints $s\sb{5- 2\ell}=0$, then in the sector $\Omega\sb \ell:{3\pi\over 5}+{2\pi\over 5}\ell<\varphi=\text{arg} x<{7\pi\over 5}+{2\pi\over 5}\ell$, $\ell=0$, $\pm 1,\dots$, the solution $y(x)$ has asymptotics $$y(x)=\mu\sum\sp \infty\sb{n=0}a\sb nu\sp{-sn}+{\cal O}(u\sp{-\infty})\text{ where } u=\vert x/6\vert\sp{1/2}\exp \left[i \left({\varphi-\pi \over 2}-\pi \ell \right)\right]\text { as } \vert x\vert\to\infty\text { in } \Omega\sb \ell.$$
[J.Schmeelk (Richmond)]
MSC 2000:
*34M55 Painlevé and other special equations
34E20 Asymptotic singular perturbations, methods (ODE)
34E05 Asymptotic expansions (ODE)
42A38 Fourier type transforms, one variable

Keywords: isomonodromy deformation method; monodromy data; first Painlevé equation; asymptotic descriptions; Stokes multiplier constraints; sectors of the complex plane

Cited in: Zbl 1024.34081

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