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

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Claeys, Tom [UCL]

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.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès libre
Publication date 2012
Language Anglais
Journal information "Random Matrices: Theory and Applications" - Vol. 1, no. 1130002, p. 1-24 (2012)
Peer reviewed yes
Publisher World Scientific
issn 2010-3263
e-issn 2010-3271
Publication status Publié
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Bibliographic reference Claeys, Tom. The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems. In: Random Matrices: Theory and Applications, Vol. 1, no. 1130002, p. 1-24 (2012)
Permanent URL http://hdl.handle.net/2078.1/136111