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Studies on the geometry of Painlevé equations

Stokes, Alexander Henry; (2020) Studies on the geometry of Painlevé equations. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis is a collection of work within the geometric framework for Painlevé equations. This approach was initiated by the Japanese school, and is based on studying Painlevé equations (differential or discrete) via certain rational surfaces associated with affine root systems. Our work is grouped into two main themes: on the one hand making use of tools and techniques from the geometric framework to study problems from applications where Painlevé equations appear, and on the other hand developing and extending the geometric framework itself. Differential and discrete Painlevé equations arise in a wide range of areas of mathematics and physics, and we present a general procedure for solving the identification problem for Painlevé equations. That is, if a differential or discrete system is suspected to be equivalent to one of Painlevé type, we outline a method, based on constructing the associated surfaces explicitly, for identifying the system with a standard example, in which case known results can be used, and demonstrate it in the case of equations appearing in the theory of orthogonal polynomials. Our results on the geometric framework itself begin with an observation of a new class of discrete equations that can described through the geometric theory, beyond those originally defined by Sakai in terms of translation symmetries of families of surfaces. To be precise, we build on previous studies of equations corresponding to non-translation symmetries of infinite order (so-called projective reductions, with fewer parameters than translations of the same surface type) and show that Sakai’s theory allows for integrable discrete equations to be constructed from any element of infinite order in the symmetry group and still have the full parameter freedom for their surface type. We then also make the first steps toward a geometric theory of delay-differential Painlevé equations by giving a description of singularity confinement in this setting in terms of mappings between jet spaces.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Studies on the geometry of Painlevé equations
Event: UCL (University College London)
Language: English
Additional information: Copyright © The Author 2020. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10109370
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