Rational Solutions of the Painlevé‐III Equation

Autor: Yue Sheng, Peter D. Miller, Thomas Bothner
Rok vydání: 2018
Předmět:
Zdroj: Bothner, T, Miller, P D & Sheng, Y 2018, ' Rational Solutions of the Painlevé-III Equation ', Studies in Applied Mathematics, pp. 626-679 . https://doi.org/10.1111/sapm.12220
Bothner, T J, Miller, P D & Sheng, Y 2018, ' Rational Solutions of the Painlevé-III Equation ', STUDIES IN APPLIED MATHEMATICS, vol. 141, no. 4, pp. 626-679 . https://doi.org/10.1111/sapm.12220
ISSN: 1467-9590
0022-2526
DOI: 10.1111/sapm.12220
Popis: All of the six Painlev´e equations except the first have families of rationalsolutions, which are frequently important in applications. The third Painlev´eequation in generic form depends on two parameters m and n, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as n ∈ Z increases and how the patterns vary with m ∈ C. This study suggests that it is reasonable to consider the rational solutions in the limit of large n ∈ Z with m ∈ C being an auxiliary parameter. To analyze the rational solutions in this limit, algebraic techniques need to be supplemented by analytical ones, and the main new contribution of this paper is to develop a Riemann–Hilbert representation of the rational solutions of Painlevé-III that is amenable to asymptotic analysis. Assuming further that m is a half-integer, we derive from the Riemann–Hilbert representation a finite dimensional Hankel system for the rational solution in which n ∈ Z appears as an explicit parameter.
Databáze: OpenAIRE
Externí odkaz:
Nepřihlášeným uživatelům se plný text nezobrazuje