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Zbl 0662.34056
Kitaev, A.V.
The method of isomonodromy deformations and asymptotics of solutions of the complete'' third PainlevĂ© equation.
(Russian)
[J] Mat. Sb., N. Ser. 134(176), No.3(11), 421-444 (1987).

Using the method of the monodromy preserving deformation, the author studies a $2\times2$ matrix linear ordinary differential equation $$\Phi\sb\lambda = \tau \left\{-i\sigma\sb3 - \frac{ai}{2\tau\lambda} \sigma\sb3 - \frac1\lambda \pmatrix 0 & C \\ D & 0 \endpmatrix + \frac{i}{2\lambda\sp2} \pmatrix \sqrt{\theta\sp2-AB} & A \\ B & -\sqrt{\theta\sp2-AB} \endpmatrix \right\} \Phi, \tag1$$ where $\sigma\sb3 = \pmatrix 1 & 0 \\ 0 & -1 \endpmatrix$, $\theta\sp2=1$ and $A = A(\tau)$, $B = B(\tau)$, $C = C(\tau)$, $D = D(\tau)$ are meromorphic functions in the region $\Re\tau>0$. He obtains asymptotic formulas for the solutions of the complete'' third PainlevĂ© equation $$u\sb{zz} = \frac{u\sp2\sb z}{u} - \frac{u\sb z}{z} + \frac{1}{z} (\hat au\sp2 u\sp2 + \hat b) + \hat c u\sp3 + \frac{\hat d}{u}, \quad \hat c\hat d\ne 0,$$ the solutions of which are connected with the monodromy preserving deformations of the coefficients of (1).
[J.Kalas]
MSC 2000:
*34E05 Asymptotic expansions (ODE)

Keywords: monodromy preserving deformation; third PainlevĂ© equation

Cited in: Zbl 0716.34073

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