2014/10/31 by Nalini Joshi · 10 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Asymptotic expansion #Domain (mathematical analysis) #Infinity #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Phase space #Physics #Power series #Quantum mechanics #Series (stratigraphy) #Space (punctuation) #Variable (mathematics)
paper · doi:10.1111/sapm.12066
published in Studies in Applied Mathematics 134(2), 233-251 (Wiley)
openalex publication_date 2014/10/31 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28
In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q ‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation ( q P I ), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q ‐domain. The method, while demonstrated for q P I , is also applicable to other q ‐difference Painlevé equations.