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On existence and properties of roots of third Painlevé' transcendents

S. I. Tertychniy

TL;DR

This work develops a rigorous framework for existence, regularity, and approximation of Third Painlevé transcendents near nonzero roots. By reformulating $P_{III}$ in a Hamiltonian setting, it yields two coupled Riccati systems whose regular solutions correspond to analytic $P_{III}$-functions vanishing at prescribed points. It then casts the problem into nonlinear integral equations for auxiliary variables, and proves the existence and convergence of analytic near-root solutions via an iterative scheme, with precise domain and bound estimates. A central result identifies the root indexing by $\mathbb{Z}_2\times\mathbb{C}$, and a root–pole correspondence is established through symmetry that maps zeros to simple poles under parameter interchanges. Finally, the paper provides explicit near-root representations (including an $8^{th}$-order-like expansion) and numerical validation, offering practical tools to compute $P_{III}$ solutions and to bound root distances and root–pole separations.

Abstract

Separate consideration of properties of roots of Third Painlevé transcendents (P_III-functions) is necessary due to irregularity the differential equation defining them reveals on the subset of the phase space where its solution would vanish. Application of the Hamiltonian formalism enables one to replace the mentioned second order differential equation (Third Painlevé equation) by two independent systems of two nonlinear first order equations whose structures allow to name them coupled Riccati equations. The existence of P_III-functions vanishing at a given non-zero point then follows, all they being analytic thereat. The set $\mathbb{Z}_2\times \mathbb{C}$ (or $\mathbb{Z}_2\times \mathbb{R}$) can be used for their indexing. It proves also to be natural to use as an unknown the third order derivative rather than the original nknown itself. After transformation of the corresponding differential equations to equivalent integral equations the efficient algorithm of the constructing of approximate solutions to Third Painlevé equation in vicinity of their non-zero root in the form of truncated power series is obtained. An example of its application is given, its numerical validation presenting results in a graphical form is carried out. The associated approximation applicable in vicinity of a pole of the corresponding P_III-function is given as well. The bounds from below for the distances between a pair of roots of a P_III-function and between a root and a pole representable in terms of elementary functions are derived.

On existence and properties of roots of third Painlevé' transcendents

TL;DR

This work develops a rigorous framework for existence, regularity, and approximation of Third Painlevé transcendents near nonzero roots. By reformulating in a Hamiltonian setting, it yields two coupled Riccati systems whose regular solutions correspond to analytic -functions vanishing at prescribed points. It then casts the problem into nonlinear integral equations for auxiliary variables, and proves the existence and convergence of analytic near-root solutions via an iterative scheme, with precise domain and bound estimates. A central result identifies the root indexing by , and a root–pole correspondence is established through symmetry that maps zeros to simple poles under parameter interchanges. Finally, the paper provides explicit near-root representations (including an -order-like expansion) and numerical validation, offering practical tools to compute solutions and to bound root distances and root–pole separations.

Abstract

Separate consideration of properties of roots of Third Painlevé transcendents (P_III-functions) is necessary due to irregularity the differential equation defining them reveals on the subset of the phase space where its solution would vanish. Application of the Hamiltonian formalism enables one to replace the mentioned second order differential equation (Third Painlevé equation) by two independent systems of two nonlinear first order equations whose structures allow to name them coupled Riccati equations. The existence of P_III-functions vanishing at a given non-zero point then follows, all they being analytic thereat. The set (or ) can be used for their indexing. It proves also to be natural to use as an unknown the third order derivative rather than the original nknown itself. After transformation of the corresponding differential equations to equivalent integral equations the efficient algorithm of the constructing of approximate solutions to Third Painlevé equation in vicinity of their non-zero root in the form of truncated power series is obtained. An example of its application is given, its numerical validation presenting results in a graphical form is carried out. The associated approximation applicable in vicinity of a pole of the corresponding P_III-function is given as well. The bounds from below for the distances between a pair of roots of a P_III-function and between a root and a pole representable in terms of elementary functions are derived.

Paper Structure

This paper contains 21 sections, 20 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Third Painlevé equation from viewpoint of the Hamiltonian formalism
  3. Some features of nonzero roots of Third Painlevé transcendents
  4. Integral equations controlling Third Painlevé transcendents near their roots
  5. Analytic solutions to integral equations for Third Painlevé transcendents and theorem \ref{['teo::020']}
  6. Iterative form of the alternative system of integral equation
  7. Initial case
  8. Algorithm of iterative constructing of approximate solutions
  9. Sequence of approximate solutions to integral equations and its convergence
  10. Bounding minimal distance between a non-zero root and other roots and poles
  11. Approximate representation of Third Painlevé transcendent near its root
  12. Derivation of explicit approximate solution
  13. Numerical example
  14. Application to poles
  15. Summary
  16. ...and 6 more sections

Key Result

Proposition 1

Let the function $\lambda$ of the variable $t$ obey the equation equation PIII' and $\upgamma,\updelta$ be arbitrary nonzero constants. Then solves Eq. $\!$equation PIII in which the rest two constant parameters are set as follows:

Figures (4)

  • Figure 1: The result of numerical integrating of the Third Painlevé equation \ref{["equation PIII'"]} on the interval $[0.01,2]$ is shown. The values of the approximate solution $\lambda(t)$ and of its argument $t$ are plotted against the vertical and horizontal directions, respectively, The constant parameters fixing the equation were set as follows: $\chi_0 = -0.811598...$, $\chi_\infty=-0.055004...$ The solution is fixed by the values of $\lambda(t)=0.288298...$ and $\dot\lambda(t)=0.374531...$ taken for $t=0.833651...$ that is indicated by the horizontal, inclined, and vertical dashed line segments, respectively, intersecting at the point of specification of the listed initial data.
  • Figure 2: The difference of the left and right hand sides of Eq. $\!$\ref{["equation PIII'"]} (the residual discrepancy) evaluated on the function $\lambda(t)$ displayed in figure \ref{['Fig1']} is shown. The values of the argument $t$ are plotted against the horizontal coordinate axis. The positions of six roots of $\lambda(t)$ are indicated by small disks situated on the latter.
  • Figure 3: The third order derivative of the function $\lambda(t)$ displayed in figure \ref{['Fig1']} is plotted on a part of its domain. The positions of the two selected roots of $\lambda(t)$ picked out therein are indicated.
  • Figure 4: Overlapping between the two graphs imaging the approximations defined by the formula \ref{['lam_6 definition']} which are centered at $t_{0_1}=0.5111...$ and $t_{0_2}=1.3818...$, respectively, (dashed lines) and the plot of numerical solution to Eq. $\!$\ref{["equation PIII'"]} vanishing at $t_{0_1}$ and $t_{0_2}$ (gray solid line in background) also displayed on a greater domain in figure \ref{['Fig1']}.

Theorems & Definitions (25)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Corollary 8
  • Corollary 9
  • Proposition 10
  • ...and 15 more