Painlevé Transcendents: The Riemann-Hilbert Approach

• Introduction. Painlevé transcendents as nonlinear special functions

Part 1. Riemannian-Hilbert problem, isomonodromy method and special functions

• Systems of linear ordinary differential equations with rational coefficients. Elements of the general theory
• Monodromy theory and special functions
• Inverse monodromy problem and Riemann-Hilbert factorization
• Isomonodromy deformations. The Painlevé equations
• The isomonodromy method
• Bäcklund transformations

Part 2. Asymptotics of the Painlevé II transcendent. A case study

• Asymptotic solutions of the second Painlevé equation in the complex plane. Direct monodromy problem approach
• Asymptotic solutions of the second Painlevé equation in the complex plane. Inverse monodromy problem approach
• PII asymptotics on the canonical six-rays. The purely imaginary case
• PII asymptotics on the canonical six-rays. Real-valued case
• PII quasi-linear Stokes phenomenon

Part 3. Asymptotics of the third Painlevé transcendent

• PIII equation, an overview
• Sine-Gordon reduction of PIII
• Canonical four-rays. Real-valued solutions of SG-PIII
• Canonical four-rays. Singular solutions of the SG-PIII
• Asymptotics in the complex plane of the SG-PIII transcendent
• Proof of Theorem 3.4
• The Birkhoff-Grothendieck theorem with a parameter
• Bibliography
• Subject index