The method of isomonodromy deformations and the asymptotics of solutions of the “complete” third Painlevé equation

A $2\times2$ matrix linear ordinary differential equation of the first order is considered whose coefficients depend on an additional parameter $\tau$ having two irregular first order singular points $\lambda=0$ and $\lambda=\infty$. The monodromy data of this equation as $\tau\to0$ and $\tau\to\infty$ are computed. These computations are used to find the asymptotics of the “degenerate” fifth Painlevé equation, which is equivalent to the “complete” third one. This is possible due to the connection of these Painlevé equations with isomonodromy deformations of the coefficients of the matrix linear equation. Bäcklund transformations and their application to asymptotic problems are considered in detail.
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