### Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

The Guest Editors for this special issue are

Percy Deift (New York University, USA)
Boris Dubrovin (SISSA, Italy)
Tamara Grava (SISSA, Italy and University of Bristol, UK)
Alexander Its (Indiana University - Purdue University Indianapolis, USA)
Peter Miller (University of Michigan, USA)

This issue is partly related to the workshop on Painlevé Equations and Applications held at the University of Michigan, August 25–29, 2017. It also serves to honor the memory of Andrei Kapaev, who made many important contributions to the field and who passed away unexpectedly in the prime of his life and the peak of his research activity.

Painlevé transcendents are often called “nonlinear special functions” or “special functions of the 21st century” for the key role they play in studying nonlinear transitional and critical phenomena in many problems of modern mathematics and theoretical physics, especially in random matrix theory, statistical mechanics and partial differential equations. The analytical basis making Painlevé functions genuine special functions is the remarkable possibility of writing down explicit connection formulae between the asymptotic parameters at the relevant critical points for all six Painlevé equations.

The vast majority of the known connection formulae for Painlevé transcendents have been obtained, via the Isomonodromy/Riemann–Hilbert technique, by Andrei Kapaev. Moreover, Andrei was one of the principal creators of the technique itself. Among Kapaev’s most celebrated results are the complete lists of the asymptotics in the complex domain and the corresponding connection formulae for the solutions of the first, second and fourth Painlevé equations, the complete description of all possible scaling limits (double asymptotics in the argument and the parameter) of the second Painlevé equation at the level of solutions, and the pioneering evaluation of the Stokes constants for the tritronquée solutions of the first and second Painlevé equations. Indeed, one can say that the global asymptotic analysis of the Painlevé equations, as we know it, has been shaped to a great extent by the outstanding work of Andrei Kapaev, to whose memory this special issue is appreciatively dedicated.

This special issue covers some of the principal directions in the modern theory and applications of Painlevé equations. We believe this is a good way to pay tribute to the memory of Andrei Kapaev. Almost all of the papers in this volume cite Kapaev’s pioneering results. Andrei left us when he had so many unfinished projects. One of them was recently completed by Boris Dubrovin, his long time co-author and co-editor of this issue, and it is presented here in its entirety. Many others are still unfinished ...

We wish to thank also all the workshop participants, all the authors who have published papers in the issue, and to give our special thanks to all the referees for providing constructive reports.

The Guest Editors

Papers in this Issue:

On the Increasing Tritronquée Solutions of the Painlevé-II Equation
Peter D. Miller
SIGMA 14 (2018), 125, 38 pages   [ abs   pdf ]
On Solutions of the Fuji-Suzuki-Tsuda System
Pavlo Gavrylenko, Nikolai Iorgov and Oleg Lisovyy
SIGMA 14 (2018), 123, 27 pages   [ abs   pdf ]
Truncated Solutions of Painlevé Equation ${\rm P}_{\rm V}$
Rodica D. Costin
SIGMA 14 (2018), 117, 14 pages   [ abs   pdf ]
Three-Parameter Solutions of the PV Schlesinger-Type Equation near the Point at Infinity and the Monodromy Data
Shun Shimomura
SIGMA 14 (2018), 113, 50 pages   [ abs   pdf ]
Large $z$ Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane
Alfredo Deaño
SIGMA 14 (2018), 107, 19 pages   [ abs   pdf ]
Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators
Victor Yu. Novokshenov
SIGMA 14 (2018), 106, 13 pages   [ abs   pdf ]
Tronquée Solutions of the Third and Fourth Painlevé Equations
Xiaoyue Xia
SIGMA 14 (2018), 095, 28 pages   [ abs   pdf ]
A Riemann-Hilbert Approach to the Heun Equation
Boris Dubrovin and Andrei Kapaev
SIGMA 14 (2018), 093, 24 pages   [ abs   pdf ]
Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
Marco Bertola, José Gustavo Elias Rebelo and Tamara Grava
SIGMA 14 (2018), 091, 34 pages   [ abs   pdf ]
Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
Galina Filipuk and Walter Van Assche
SIGMA 14 (2018), 088, 19 pages   [ abs   pdf ]
Notes on Non-Generic Isomonodromy Deformations
Davide Guzzetti
SIGMA 14 (2018), 087, 34 pages   [ abs   pdf ]
The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
Mattia Cafasso and Manuel D. de la Iglesia
SIGMA 14 (2018), 076, 20 pages   [ abs   pdf ]
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations
Anton Dzhamay and Tomoyuki Takenawa
SIGMA 14 (2018), 075, 20 pages   [ abs   pdf ]
Numerical Approach to Painlevé Transcendents on Unbounded Domains
Christian Klein and Nikola Stoilov
SIGMA 14 (2018), 068, 10 pages   [ abs   pdf ]
Poles of Painlevé IV Rationals and their Distribution
Davide Masoero and Pieter Roffelsen
SIGMA 14 (2018), 002, 49 pages   [ abs   pdf ]