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Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations

Malik Balogoun, Marco Bertola

TL;DR

This work connects rational solutions of $P_V$ to Hankel determinants formed from semiclassical moments, casting the tau function as an isomonodromic tau function and embedding it in a Riemann–Hilbert framework. Using the Deift–Zhou steepest descent method, the authors construct a genus-zero $g$-function outside an almond-shaped region ${\rm EoT}$ and a genus-one $g$-function inside ${\rm EoT}$ to analyze large-degree tau functions. They show that zeros of the scaled tau function ${\mathcal T}_n(s)$ concentrate in ${\rm EoT}$, structured by a pair of quantization conditions that yield a semi-regular lattice, with explicit model solutions expressed via elliptic theta functions. Outside ${\rm EoT}$, the RH problem is solvable for large $n$, placing poles (zeros) of the PV rational solutions inside a neighborhood of ${\rm EoT}$; inside ${\rm EoT}$ the zeros align with a refined elliptic mesh, captured by a theta-function-based model. The results provide a precise, quantifiable description of pole locations and illuminate the asymptotic geometry of PV rational solutions in the large-degree limit.

Abstract

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.

Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations

TL;DR

This work connects rational solutions of to Hankel determinants formed from semiclassical moments, casting the tau function as an isomonodromic tau function and embedding it in a Riemann–Hilbert framework. Using the Deift–Zhou steepest descent method, the authors construct a genus-zero -function outside an almond-shaped region and a genus-one -function inside to analyze large-degree tau functions. They show that zeros of the scaled tau function concentrate in , structured by a pair of quantization conditions that yield a semi-regular lattice, with explicit model solutions expressed via elliptic theta functions. Outside , the RH problem is solvable for large , placing poles (zeros) of the PV rational solutions inside a neighborhood of ; inside the zeros align with a refined elliptic mesh, captured by a theta-function-based model. The results provide a precise, quantifiable description of pole locations and illuminate the asymptotic geometry of PV rational solutions in the large-degree limit.

Abstract

In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.

Paper Structure

This paper contains 20 sections, 18 theorems, 169 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Semiclassical orthogonal polynomials and tau functions of Painlevé type
  3. Special case of semiclassical functional of PV type
  4. From orthogonal polynomials to Lax pairs
  5. Asymptotic analysis
  6. Construction of the $\boldsymbol{g}$-function
  7. Outside $\boldsymbol{{\rm EoT}}$: Genus $\boldsymbol{0}$ $\boldsymbol{g}$-function and effective potential
  8. Inside $\boldsymbol{{\rm EoT}}$: Genus one $\boldsymbol{g}$-function and effective potential
  9. Deift--Zhou steepest descent analysis
  10. Asymptotic analysis for $\boldsymbol{s}$ outside $\boldsymbol{{\rm EoT}}$
  11. Conclusion of the steepest descent analysis
  12. Asymptotic analysis inside $\boldsymbol{{\rm EoT}}$
  13. Second transformation, model problem and its solution: The case $\boldsymbol{\operatorname{Re} (s)>0}$
  14. Second transformation, model problem and its solution: The case $\boldsymbol{\operatorname{Re} (s)<0}$
  15. Local parametrices
  16. ...and 5 more sections

Key Result

Theorem 1.1

The equation $P_5\bigl(\alpha,\beta,\gamma,-\frac{1}{2}\bigr)$PVrat admits rational solutions if and only if there are integers $k,m\in {\mathbb Z}$ such that In cases $(I)$ and $(II)$, the solution is unique if $\gamma \not\in{\mathbb Z}$ and there are at most two rational solutions otherwise.

Figures (12)

  • Figure 1: The zeros of several instances of polynomial tau functions (all for ${K}=0$).
  • Figure 2: Some plot of the zeros of the tau function $\mathcal{T}_n(s)$\ref{['Pns']} for $(n,\rho) = (30,1), (30,2), (30,3)$ ($K=0$ in all cases). Observe that for $\rho$ integer our description of the distribution of zeros does not apply and in particular the region ${\rm EoT}$ remains largely empty, with an "eyelash" effect. There is a high multiplicity zero at the origin. As $n$ increases but $\rho$ remains fixed, the "eyelashes" become thinner around the edge of ${\rm EoT}$. A similar phenomenon was observed, for rational solutions of PIII, in Ref1_1. We are not going to discuss the case $\rho\in {\mathbb Z}$ in this paper. Clearly the behaviour of the zeros undergoes a substantial change, as they appear to be accumulating along part of the boundary of ${\rm EoT}$.
  • Figure 3: From top to bottom, left to right: $(n,{ {\rho}}) = (16,\frac{ 3}{100} + \frac{13{\rm i}}{100}), (16,\frac{ 101}{100} + \frac{13{\rm i}}{100}), (17,3+1{\rm i}), (17, {\frac{1}{2} +\frac{\rm i}{2}}), (26, \frac{101}{100}), (40, \frac{3}{100})$. Observe that the accuracy of the localization of the zeros by the grid depends on the ratio $|{ {\rho}}|/n$, the smaller the better the approximation. For example, in the case in the top right $(n,{ {\rho}}) = (17,3+ 1{\rm i})$ the ratio is significant (approximately $|{ {\rho}}|/n \simeq 0.19$) and an analysis where ${ {\rho}}$ is considered as scaling would undoubtedly be more appropriate. In all cases, we have ${K}=0$. See the explanation in Remark \ref{['remexp2']}.
  • Figure 4: Left: the level sets of $\operatorname{Re} \varphi_0(z)=0$. Right: the boundary of the region of validity of the genus-zero assumption (ignore the vertical rays issuing from $\pm 2{\rm i}$). It is the locus of $\operatorname{Re} \varphi_0(1/s)=0$. The inside of this region we refer to as the "Eye of the Tiger" (${\rm EoT}$).
  • Figure 5: The contours $\Gamma_m$, $\Gamma_c$ and the regions where $\operatorname{Re} \varphi(z;s)<0$ (shaded). Indicated also the boundaries, $\mathscr L_\pm$, of the lens regions $\Lambda_\pm$.
  • ...and 7 more figures

Theorems & Definitions (28)

  • Theorem 1.1: Kitaev94
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2: comparison with ClarksonDunning
  • Remark 2.3
  • Theorem 2.5: FIK
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • ...and 18 more