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Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

Robert J. Buckingham, Peter D. Miller

Abstract

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstrass equation.

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

Abstract

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstrass equation.
Paper Structure (19 sections, 2 theorems, 109 equations, 3 figures)

This paper contains 19 sections, 2 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebraic solutions of the Painlevé-III ($\boldsymbol{{\rm D}_7}$) equation
  3. Formal degeneration of Painlevé-III (D$\boldsymbol{_7}$)
  4. Riemann--Hilbert representation of $\boldsymbol{U_n(y):=n^{-1/2}u_n(n^{3/2}y)}$
  5. Main aims of the paper
  6. Spectral curves of genus 1
  7. Introduction of $\boldsymbol{g}$-function and lens opening
  8. First step: introduction of $\boldsymbol{g}$-function
  9. Second step: opening a lens
  10. Outer parametrix Riemann--Hilbert problem
  11. Derivation of the Weierstraß differential equation for $\boldsymbol{\breve{U}_n(z;y)}$
  12. Expansion of $\boldsymbol{\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)}$ near $\boldsymbol{\eta=\infty}$
  13. Differential equations in $\boldsymbol{z}$
  14. Lax equation satisfied by $\boldsymbol{\mathbf{F}^{(n)}(\eta,y,z)}$
  15. Implied differential equations for coefficients
  16. ...and 4 more sections

Key Result

Lemma 2.1

Assume that $\operatorname{Re}(y)>0$ and that $Y=y^\frac{1}{3}$ lies in the open interior of $\mathcal{B}$. Then, there is a well-defined value $c=c_1(y)\in\mathbb{C}$, a smooth function of real variables $\operatorname{Re}(y)$ and $\operatorname{Im}(y)$ but not analytic in $y$, such that the follow

Figures (3)

  • Figure 1: Left panel: a density plot of $|U_{10}\bigl(Y^3\bigr)|$ and the boundary of the "bow-tie" region $\mathcal{B}$. Right panel: a similar plot of $|U_{10}(y)|$ on the principal sheet of the $y$-plane with $-\pi<\arg(y)<\pi$ and the sheet boundary (branch cut) shown with a red line. In both plots, lighter/darker color indicates larger/smaller modulus.
  • Figure 2: The jump contour for Riemann--Hilbert Problem \ref{['rhp:Z']}.
  • Figure 3: Jump contours and sign chart of $\operatorname{Re}(h)$ in the $\eta$-plane for $0<y<y_\mathrm{c}$. Left panel: the zero level set $K$ of $\operatorname{Re}(h(\eta,y,c))$ shown in gray and orange (orange indicates the branch cuts $\Sigma_{0,1}$ and $\Sigma_{2,3}$ of $h_\eta(\eta,y,c)$), and the relative placement of the jump contour for Riemann--Hilbert Problem \ref{['rhp:Z']}. The sign of $\operatorname{Re}(h)$ is as indicated and $\operatorname{Re}(h)$ only changes sign across the gray arcs of $K$. Note that the contour $\Sigma_\infty^+$ actually extends from $\eta=\mathrm{i} s_2$, taken as the junction point of $C^-$, $C^+$, and $\Sigma_0^+$, all the way up the positive imaginary axis, passing through $\Sigma_{2,3}$. Likewise $\Sigma_\infty^-$ extends from $-\mathrm{i}\infty$ up to $\mathrm{i} s_1$. The arc $\Sigma_0^-$ coincides with the branch cut $\Sigma_{0,1}$. Center panel: the jump contour for $\mathbf{N}^{(n)}(\eta,y,z)$ has two additional arcs on the left and right of the branch cut $\Sigma_{2,3}$ after opening a lens. Right panel: the jump contour for $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ consists of the arcs $\Sigma_{\infty}^-$, $\Sigma_{0,1}$, $\Sigma_0^+$, and $\Sigma_{2,3}$ (shown with solid curves; the dashed arcs in the jump contour for $\mathbf{N}^{(n)}(\eta,y,z)$ have been neglected).

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 5.1
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4
  • Remark 5.5