Alamoudi, Khadija; (2018) The use of singularity structure to find special solutions of differential equations: an approach from Nevanlinna theory. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

For a certain family of ordinary differential equations, Nevanlinna theory is used to find all solutions in a special class. A differential equation is said to possess the (strong) Painlev´e property if all solutions have only poles as movable singularities. The solutions of such equations are particularly well behaved and the Painlev´e property is closely associated with integrability. In this thesis, we extend the idea of using singularity structure to find all special solutions with good singularity structure, even when the general solution is badly behaved. We begin by finding solutions that are meromorphic in the complex plane and more complicated than the coefficients in the equation in a sense made precise by Nevanlinna theory. Such meromorphic solutions are called admissible and include all non-rational meromorphic solutions of an equation with rational coefficients. The use of Nevanlinna theory in the entire complex plane does not allow the solutions to be branched at fixed singularities, which seems more natural from the perspective of the Painlev´e property. Motivated by this, we consider an extension of Nevanlinna theory to a large sector-like region with a deleted disc to allow for such branching and apply this theory to differential equations.

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