The Dirichlet Boundary Value Problem for Real Solutions of the first Painlev\'e Equation on Segments on Non-Positive Semi-Axis

N. Joshi; A. V. Kitaev

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlev\'e Equation on Segments on Non-Positive Semi-Axis

Classical Analysis and ODEs 2007-05-23 v1 Mathematical Physics math.MP Exactly Solvable and Integrable Systems

Authors: N. Joshi , A. V. Kitaev

Abstract

We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$ . In this work a restriction $x\leq0$ 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 $y(a)=y^0$ , $y(b)=y_0$ for $a<b\leq0$ .

Keywords

first Painlevé equation real solutions Dirichlet boundary value problem transcendental functions qualitative theory non-positive semi-axis

Cite

@article{arxiv.math/0407432,
  title  = {The Dirichlet Boundary Value Problem for Real Solutions of the first Painlev\'e Equation on Segments on Non-Positive Semi-Axis},
  author = {N. Joshi and A. V. Kitaev},
  journal= {arXiv preprint arXiv:math/0407432},
  year   = {2007}
}

Comments

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

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlev\'e Equation on Segments on Non-Positive Semi-Axis

Abstract

Keywords

Cite

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