Connection problem of the first Painlevé transcendents with large initial data
| Autor: | Wen-Gao Long, Yu-Tian Li |
|---|---|
| Rok vydání: | 2023 |
| Předmět: |
Statistics and Probability
Nonlinear Sciences - Exactly Solvable and Integrable Systems Mathematics - Complex Variables 33E17 34M55 41A60 FOS: Physical sciences General Physics and Astronomy Statistical and Nonlinear Physics Mathematics - Classical Analysis and ODEs Modeling and Simulation Classical Analysis and ODEs (math.CA) FOS: Mathematics Exactly Solvable and Integrable Systems (nlin.SI) Complex Variables (math.CV) Mathematical Physics |
| Zdroj: | Journal of Physics A: Mathematical and Theoretical. 56:175201 |
| ISSN: | 1751-8121 1751-8113 |
| DOI: | 10.1088/1751-8121/acc620 |
| Popis: | In previous work, Bender and Komijani (2015 \textit{J. Phys. A: Math. Theor.} 48, 475202) studied the first Painlev\'e (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a $\mathcal{PT}$-symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions $(y(0),y'(0))=(a,b)$ located on a sequence of curves $\Gamma_n$, $n=1,2,\dots$, will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation $b^2/4 - a^3=f_n \sim A n^{6/5}$ for the curves $\Gamma_{n}$ as $n\to\infty$ is derived, where $A$ is a positive constant. The discrete set $\{f_n\}$ could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of $\Gamma_n$ matches the numerical results remarkably well, even for small $n$. The main tool is the method of uniform asymptotics introduced by Bassom et al. (1998 \textit{Arch. Rational Mech. Anal.} {143}, 241--271) in the studies of the second Painlev\'e equation. Comment: 27 pages, 4 figures |
| Databáze: | OpenAIRE |
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