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Connection formulae for asymptotics of the fifth Painlevé transcendent on the imaginary axis. I. (English) Zbl 1470.34237

The main subject is to study asymptotic expansions for general solutions to the fifth Painlevé equation \[ y^{\prime\prime} = \left(\frac{1}{2y}+\frac{1}{y-1} \right) \left( \frac{dy}{dt} \right)^2-\frac{1}{t} \frac{dy}{dt} +\frac{(y-1)^2}{t^2}\left(\alpha y +\frac{\beta}{y}\right) +\gamma {\frac y t}+\delta \frac{y(y+1)}{y-1}, \] when \(t \to \pm i\, \infty\). The asymptotics are parameterized by monodromy data and the Stokes data of the associated \(2 \times 2\)-linear differential equation \[ \frac{dY }{d \lambda} = \left( \frac t 2 \sigma_3 +\frac{A_0}\lambda +\frac{A_1}{\lambda -1} \right) Y, \] where \(\sigma_3= \begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix} \). The complex parameters of the fifth Painlevé equation are described by the local exponents of each singular point of the linear differential equation. The asymptotics \(t\to \infty\) when \(t\) is on the real axis is studied by the authors [Nonlinearity 13, No. 5, 1801–1840 (2000; Zbl 0970.34076)]
The main result is the asymptotic expansions of generic solutions to the fifth Painlevé equation in two complex domains. Each domain consists of two semi-infinite strips containing the positive and negative imaginary semiaxis without some neighborhood of the origin and has a countable number of holes.
The monodromy data corresponding to each Painlevé function is represented by the local exponents and parameters of asymptotic expansion. To obtain the monodromy data, we apply the WKB analysis around \(\lambda=0\) and \(\lambda=1\) and match two solutions in the leading term.
The asymptotic expansions of the fifth Painlevé equation in the case \(t \to +\infty\) are studied by the authors [loc. cit.]. Comparison with the result by B. M. McCoy and S. Tang [Physica D 19, 42–72 (1986; Zbl 0639.58041)] is also described. Numerical verifications and graphs of numerical solutions are given in the last part.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
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