We study the application of Bose-Einstein condensates (BECs) to simulations of phenomena across a number of disciplines in physics, using theoretical and computational methods. ¶ Collapsing condensates as created by E. Donley et al. [Nature 415, 39 (2002)] exhibit potentially useful parallels to an inflationary universe. To enable the exploitation of this analogy, we check if current quantum field theories describe collapsing condensates quantitatively, by targeting the discrepancy between experimental and theoretical values for the time to collapse. To this end, we couple the lowest order quantum field correlation functions to the condensate wavefunction, and solve the resulting Hartree-Fock-Bogoliubov equations numerically. Complementarily, we perform stochastic truncated Wigner simulations of the collapse. Both methods also allow us to study finite temperature effects. ¶ We find with neither method that quantum corrections lead to a faster collapse than is predicted by Gross-Pitaevskii theory. We conclude that the discrepancy between the experimental and theoretical values of the collapse time cannot be explained by Gaussian quantum fluctuations or finite temperature effects. Further studies are thus required before the full analogue cosmology potential of collapsing condensates can be utilised. ¶ As the next project, we find experimental parameter regimes in which stable three-dimensional Skyrmions can exist in a condensate. We show that their stability in a harmonic trap depends critically on scattering lengths, atom numbers, trap rotation and trap anisotropy. In particular, for the Rb87 |F=1,m_f=-1>, |F=2,m_f=1> hyperfine states, stability is sensitive to the scattering lengths at the 2% level. We find stable Skyrmions with slightly more than 2*10^6 atoms, which can be stabilised against drifting out of the trap by laser pinning. ¶ As a stepping stone towards Skyrmions, we propose a method for the stabilisation of a stack of parallel vortex rings in a Bose-Einstein condensate. The method makes use of a ``hollow'' laser beam containing an optical vortex, which realises an optical tunnel for the condensate. Using realistic experimental parameters, we demonstrate numerically that our method can stabilise up to 9 vortex rings. ¶ Finally, we focus on analogue gravity, further exploiting the analogy between flowing condensates and general relativistic curved space time. We compare several realistic setups, investigating their suitability for the observation of analogue Hawking radiation. We link our proposal of stable ring flows to analogue gravity, by studying supersonic flows in the optical tunnel. We show that long-living immobile condensate solitons generated in the tunnel exhibit sonic horizons, and discuss whether these could be employed to study extreme cases in analogue gravity. ¶ Beyond these, our survey indicates that for conventional analogue Hawking radiation, simple outflow from a condensate reservoir, in effectively one dimension, has the best properties. We show with three dimensional simulations that stable sonic horizons exist under realistic conditions. However, we highlight that three-body losses impose limitations on the achievable analogue Hawking temperatures. These limitations vary between the atomic species and favour light atoms. ¶ Our results indicate that Bose-Einstein condensates will soon be useful for interdisciplinary studies by analogy, but also show that the experiments will be challenging.

This thesis is concerned with various methods for the control and investigation of pulse dynamics in a Photonic Crystal Fibre (PCF) and of the radiation driven by a short pulse. In particular the focus is on pulses in the anomalous dispersion region which would form solitons in the absence of higher order effects. Several different types of radiation can be driven by such pulses if they are perturbed by higher order dispersive and non-linear effects - for example Resonant Radiation (RR) and Negative Resonant Radiation (NRR) two dispersive waves which gain energy at the expense of the pulse. The feature of NRR which is of particular importance is that it is the first observed example of a coupling between positive and negative frequencies in optics. This has only been possible due to recent advances in fields such as PCFs, lasers and analogue systems. As with many scientific discoveries, NRR was found by bringing together ideas and techniques from these different fields. Both the pulse and the driven radiation are investigated using a number of different pulse and PCF parameters. These include power, chirp, polarisation and PCF dispersion. These are used to vary the wavelengths at which the driven radiation occurs as well as its generation efficiency. Furthermore the power and chirp are used to vary where in the PCF the driven radiation is generated by controlling where the driving pulse compresses and spectrally expands. This property is used to investigate different stages in the evolution of the pulse and driven radiation as well as to optimise the generation efficiency of the driven radiation.