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Construction et étude des solutions de l'équation de Painlevé II

Waltzing, Simon
(2020)

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Details

Supervisors
Claeys, Tom
Faculty
Faculté des sciences
Degree label
Master [120] en sciences mathématiques
Abstract
enThe aim of this thesis is to build solutions for the second equation of Painlevé. After a historical introduction, the first chapter focuses on the particular case where the constant of the equation is equal to zero. To build a solution, we use a formulation by a Riemann-Hilbert problem. It also allows us to know the asymptotic behaviour of this solution. The second chapter deals with the cases of the equation where the constant is an entire number. To build a solution, we use a formulation in a Hamiltonian system and the polynomials of Yablonsky-Vorob'ev. The third and final chapter focuses on two applications of Painlevé's second equation: the link to the mKdV equation and the definition of Tracy-Widom's distribution.