|
Zapiski Nauchnykh Seminarov POMI, 2023, Volume 520, Pages 189–226
(Mi znsl7318)
|
|
|
|
One-parameter meromorphic solution of the degenerate third Painlevé equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin
A. V. Kitaeva, A. Vartanianb a Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
b Department of Mathematics, College of Charleston, Charleston, SC 29424, USA
Abstract:
We prove that there exists a one-parameter family of meromorphic solutions $u(\tau)$ vanishing at $\tau=0$ of the degenerate third Painlevé 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)},\ \varepsilon=\pm1,\ \varepsilon b>0, \end{equation*} for formal monodromy parameter $a=\pm\mathrm{i}/2$. 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.
Key words and phrases:
Painlevé equation, monodromy data, asymptotics, content of polynomial.
Received: 26.05.2023
Citation:
A. V. Kitaev, A. Vartanian, “One-parameter meromorphic solution of the degenerate third Painlevé equation with formal monodromy parameter $a=\pm\mathrm{i}/2$ vanishing at the origin”, Questions of quantum field theory and statistical physics. Part 29, Zap. Nauchn. Sem. POMI, 520, POMI, St. Petersburg, 2023, 189–226
Linking options:
https://www.mathnet.ru/eng/znsl7318 https://www.mathnet.ru/eng/znsl/v520/p189
|
Statistics & downloads: |
Abstract page: | 41 | Full-text PDF : | 25 | References: | 23 |
|