The subject of this work is a theoretical analysis of the Pearcey function. In optics, thin lens theory supposes that all rays focus at a unique point where the field converges. For a real lens, the focal point is replaced by a cusp, which is the end point of a caustic curve dividing the bright field region from the dark. My particular interest is the pattern of nodal points within the cusp. By investigating the stationary points for the cusp catastrophe, asymptotic approximations are found for the Pearcey function. This in turn leads to the development of finding the positions of nodal points inside, and outside a caustic. Also values for $|P|$ on a small circle surrounding a node are examined and show reasonable accuracy of order $10^{-8}$. / Thesis / Master of Science (MSc) / Identifying the nodes of the Pearcey function.

Masters thesis of Liam Farrell, under the supervision of Dr. Duncan O'Dell. Successfully defended on August 26, 2021. / Caustics are regions created by the natural focusing of waves. Some examples include rainbows, spherical aberration, and sonic booms. The intensity of a caustic is singular in the classical ray theory, but can be smoothed out by taking into account the interference of waves. Caustics are generic in nature and are universally described by the mathematical theory known as catastrophe theory, which has successfully been applied to physically describe a wide variety of phenomena. Interestingly, caustics can exist in quantum mechanical systems in the form of phase singularities. Since phase is such a central concept in wave theory, this heralds the breakdown of the wave description of quantum mechanics and is in fact an example of a quantum catastrophe. Similarly to classical catastrophes, quantum catastrophes require some previously ignored property or degree of freedom to be taken into account in order to smooth the phase divergence. Different forms of spontaneous pair-production appear to suffer logarithmic phase singularities, specifically Hawking radiation from gravitational black holes. This is known as the trans-Planckian problem. We will investigate Hawking radiation formed in an analogue black hole consisting of a flowing ultra-cold Bose-Einstein condensate. By moving from an approximate hydrodynamical continuum description to a quantum mechanical discrete theory, the phase singularity is cured. We describe this process, and make connections to a new theory of logarithmic catastrophes. We show that our analogue Hawking radiation is mathematically described by a logarithmic Airy catastrophe, which further establishes the plausibility of pair-production being a quantum catastrophe / Thesis / Master of Science (MSc)