Benincasa, Gregorio Benedetto; (2018) The anti-self-dual Yang-Mills equations and discrete integrable systems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this dissertation the Bäcklund-Darboux transformations for the anti-self-dual Yang-Mills (ASDYM) equations and implications of such constructions are studied. After introducing Bäcklund and Darboux type transformations and the anti-self-dual Yang-Mills equations, which are the central objects we are concerned with, two principal themes arising from these are treated. Firstly, we construct a Bäcklund-Darboux transformation for the ASDYM equations and present reductions of this transformation to the transformations of integrable sub-systems embedded in the anti-self-duality equations. We further show how the geometry of the ASDYM equations may be exploited to give a more geometric understanding of the degeneration process involved in mapping one Painlevé equation to another. Our transformation inherits some of this geometry and we exploit this feature to lift the degenerations to the transformation itself. The second theme deals with a reinterpretation of such structure. We employ the transformation for the construction of a discrete equation governing the evolution of solutions to the ASDYM equations on the lattice. This system is a lattice gauge theory defined over Z2 and we discuss the properties of such system, including some reductions and continuous limits. A Darboux transformation for this system and an extension of this system to three dimensions is also presented. We conclude with an analysis of the singular structure of the ASDYM equations.

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