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arXiv · 2305.17278

One-Parameter Meromorphic Solution of the Degenerate Third Painlev\'{e} Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin

A. V. Kitaev·A. Vartanian
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Abstract

We prove that there exists a one-parameter meromorphic solution $u(\tau)$ vanishing at $\tau=0$ of the degenerate third Painlev\'e equation, \begin{equation*} u^{\prime \prime}(\tau) \! = \! \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} \! - \! \frac{u^{\prime}(\tau)}{\tau} \! + \! \frac{1}{\tau} \! \left(-8 \varepsilon (u(\tau))^{2} \! + \! 2ab \right) \! + \! \frac{b^{2}}{u(\tau)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{equation*} for formal monodromy parameter $a=\pm i/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau)$ at $\tau=0$ and its asymptotic behaviour as $\tau\to+\infty$. These asymptotics are visualized for generic initial data.

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BibTeXRIS

A. V. Kitaev, A. Vartanian. 2023-05-26. One-Parameter Meromorphic Solution of the Degenerate Third Painlev\'{e} Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin. https://arxiv.org/abs/2305.17278

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