Riemann-Hilbert techniques and Painlevé equations

Dr. Andrej Kapaev

Number of cycles: 1

First semester: Mon 11-12:30, Tue 9-10:30 (starting from November 7)


  1. Linear differential systems.
    • Monodromy data.
    • Isomonodromic deformations.
  2. Transformations of the linear systems.
    • Schlesinger transformations.
    • Middle convolution.
    • Harnad's duality.
  3. Direct and inverse monodromy probem.
    • Painlevé equations.
    • Riemann-Hilbert (RH) boundary value problem.
    • Uniqueness/nonuniqueness of the solution to the RH problem.
  4. Existence of the RH problem solution.
    • Fredholm theory of singular integral operators approach.
    • Birkhoff-Grothendieck approach.
  5. Asymptotic solvability of the RH problem.
    • Steepest descent analysis.
    • Boutroux equations.
  6. Asymptotic analysis of the second Painlevé equation (PII) via the RH problem.
    • Regular generic asymptotics on the real line.
    • Regular degenerate asymptotics and quasi-linear Stokes phenomenon.
    • Singular asymptotics on the real line.
    • Elliptic asymptotics in a complex sector and the nonlinear Stokes phenomenon.
  7. Coefficient asymptotics in a degenerate solution of PII.
    • Linear equations with Painlevé potential and the induced Stokes phenomenon.
  8. Scaling limits in isomonodromy systems.
    • n-large asymptotics of the polynomials orthogonal on the line w.r.t. the exponential weight with a quartic potential.
  9. Integrable integral operators and TBA.