Open Problems for Painleve Equations — Problem 1

Definitions and mathematical statement

The source concerns numerical methods for the Painlevé equations, regarded there as integrable ordinary differential equations. Here an isomonodromic formulation means an associated linear differential system whose generalized monodromy data remain fixed under deformation; its associated Riemann--Hilbert formulation characterizes a solution through an analytic boundary-value problem. An integrable discretization means a discrete equation retaining an appropriate integrable structure and approaching the associated differential equation in a suitable continuum limit. The source does not specify the integrable structure or the sense of this limit.

Research problem (Clarkson, 2019, Problem 1). Investigate the following three related questions.

1. Can numerical methods exploiting integrability improve on standard Runge--Kutta methods and their variants for solving Painlevé equations?
2. How can the isomonodromic method, through an associated Riemann--Hilbert problem, be used to develop software for numerical study of Painlevé equations?
3. Should integrable discretizations be used for numerical study of Painlevé equations, given the existence, asserted in the source, of discrete Painlevé equations tending to associated continuous Painlevé equations in appropriate continuum limits?

These are methodological research questions rather than a specified theorem or quantified numerical bound. The supplied record does not select a Painlevé equation, parameter values, real or complex independent-variable domain, initial or boundary data, or treatment of singularities. It also does not define what constitutes an improvement over Runge--Kutta methods, prescribe a Riemann--Hilbert formulation, or select a discrete equation and continuum scaling. Consequently, no universal performance claim or particular convergence assertion is specified by this draft; a precise numerical task requires these choices to be stated explicitly.

Short English statement

Clarkson asks whether integrability can improve numerical computation of Painlevé equations, how associated Riemann--Hilbert problems can inform numerical software, and whether integrable discrete Painlevé equations should serve as discretizations. The source leaves the equations, data, domains, comparison criteria, and continuum limits unspecified.

Sources

• Local UnsolvedMath registry, AMR Open Problem Lists: permanent ID $801047$, exact source problem code catalogue entry, title Open Problems for Painleve Equations --- Problem 1. The supplied record attributes the problem to Clarkson, with proposed year 2019, and labels its status open; this label is supplied curation, not an independently verified literature assessment.
• Clarkson, Open Problems for Painlevé Equations (2019), Problem 1. Supplied source URL: https://arxiv.org/abs/1901.10122. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.