The Dirichlet Boundary Value Problem for Real Solutions of the first Painlevé Equation on Segments on Non-Positive Semi-Axis

2004/07/26 by N. Joshi, A. V. Kitaev, Joshi, N. +1 · 1 citation
Mathematics · Physics and Astronomy · #33E17 #34B15 #34M55 #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:33E17 #msc:34B15 #msc:34M55 #nlin.SI

paper · pdf · doi:10.48550/arxiv.math/0407432

To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

arxiv created 2004/07/26 · arxiv updated 2009/12/01

Abstract

We develop a qualitative theory for real solutions of the equation y''=6y² -x. In this work a restriction x≤0 is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem y(a)=y⁰, y(b)=y₀ for a<b≤0.

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• Classification of the real Painlevé I transcendents by zeros and connection problem: an asymptotic study

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