Cheng, Jianzhi; (2022) Singularity Structure in Detecting Differential and Discrete Integrability. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Singularity analysis plays a critical role in detecting both differential and discrete integrable systems. In the differential setting, the Painlevé property is a powerful integrability criterion. The Painlevé test is used to detect this property, and can be easily applied to both ordinary and partial differential systems. Whereas in the study of discrete equations, although there exists the singularity confinement method analogue to the Painlevé test of differential equations, it does not impose restrictive enough conditions on discrete systems. Instead, the degree growth of discrete systems is a better integrability indicator, which assesses how fast the degree of each iterate grows as a rational function of initial conditions. In the first part of the thesis, we look at the application of Painlevé analysis in general relativity. In particular, the Painlevé ODE test is applied to the Einstein field equations for the generalised Buchdahl $n=1$ polytrope to determine the equation of state when the system is integrable, and the integrable case is subsequently solved. An analysis of the Ernst equation using the Painlevé PDE test is also conducted. In the second part, the second order discrete equation y_{n−1} + y_{n−1} = (a_n + b_n y_n) / (1−y²) is studied. Using the method based on singularity confinement introduced by Halburd, the exact degree of each iterate y_n can be easily calculated. We apply this method to find special solutions. Initial conditions are classified based on the degree growth of the solution, and integrable subsystems are identified.

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