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Singularities of the First Painlev{é} Transcendent

George F. Corliss

Abstract

Consider the solution $y(t)$ for the ordinary differential equation $y' = f(t, y)$ with $t$ complex. Second-order nonlinear differential equations often exhibit patterns in their poles, branch points, and essential singularities, explored by \Pain and colleagues, 1888--1915. A variant of the ratio test applied to the Taylor series for the solution $y$ estimates the locations and orders of singularities in the First Painlev{é} Transcendent as an example. Can you suggest applications in which our singularity location analysis can provide useful insights?

Singularities of the First Painlev{é} Transcendent

Abstract

Consider the solution for the ordinary differential equation with complex. Second-order nonlinear differential equations often exhibit patterns in their poles, branch points, and essential singularities, explored by \Pain and colleagues, 1888--1915. A variant of the ratio test applied to the Taylor series for the solution estimates the locations and orders of singularities in the First Painlev{é} Transcendent as an example. Can you suggest applications in which our singularity location analysis can provide useful insights?
Paper Structure (10 sections, 12 equations, 18 figures, 1 algorithm)

This paper contains 10 sections, 12 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Time is complex?
  2. Complex singularities and series radius of convergence
  3. Complex path of integration
  4. Singularities of the First Painlevé Transcendent
  5. Locating boundary rows and columns of singularities
  6. Are locations reproducible?
  7. Closed loop path of integration
  8. Qualitative solution surface
  9. What is the value of the solution at $t = 20$?
  10. Conclusions

Figures (18)

  • Figure 1: Singularity locations for First Painlevé Transcendent with $y(0) = 0.5$, $y'(0) = 0.9$.
  • Figure 2: Real singularities for the First Painlevé Transcendent. Upper panel shows the path of integration and the locations of real-valued singularities. Lower panel shows real and imaginary components of the solution along the path of integration.
  • Figure 3: Upper panel shows the path of integration and the locations of singularities forming the upper edge of Region 1. Lower panel shows real and imaginary components of the solution along the path of integration.
  • Figure 4: Detail of Figure \ref{['fig:PainleveRegion1B']} near $t = {10} + {7}\, i$ showing details of location of two singularities. Location estimates show excellent agreement.
  • Figure 5: Upper panel shows previously located singularities and the path of integration to discover several singularities in the interior of Region 3. Lower panel shows real and imaginary components of the solution along the path of integration.
  • ...and 13 more figures