The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems

TL;DR

This paper reviews how the Riemann-Hilbert method can be used to analyze critical asymptotics in Hamiltonian perturbations of hyperbolic and elliptic PDEs, emphasizing the universality of Painlevé transcendents.

Contribution

It provides an overview of recent advances in applying the Riemann-Hilbert approach to describe universal critical behaviors in integrable PDEs.

Findings

Painlevé transcendents describe critical behavior universally.

The Riemann-Hilbert method yields rigorous asymptotic results.

Examples include well-known integrable equations.

Abstract

The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior can be described in terms of distinguished Painlev\'e transcendents, which are universal in the sense that they are, to some extent, independent of the equation and the initial data. We will consider several examples of well-known integrable equations that are expected to show this type of Painlev\'e behavior near critical points. The Riemann-Hilbert method is a useful tool to obtain rigorous results for such equations. We will explain the main lines of this method and we will discuss the universality conjecture from a Riemann-Hilbert point of view.