Solitons in higher dimensions

Project description

The localisation of energy and its transport is of great physical interest in many applications. The mechanisms by which this occurs have been widely studied in one-dimensional systems; however, in two- and three-dimensional systems a greater variety of waves and wave phenomena can be observed; for example, waves can be localised in one or both directions.

This project will start with an analysis of the nonlinear Schrodinger equation (NLS) in higher space dimensions, and with more general nonlinearities (that is, not just $\gamma=1$). Current interest in the Bose-Einstein Condensates which are being investigated in the School of Physics and Astronomy at Nottingham makes this topic particularly timely and relevant.

The NLS equation also arises in the study of astrophysical gas clouds, and in the reduction of other nonlinear wave equations using small amplitude asymptotic expansions. For example, the reduction of the equations of motion for atoms in a crystal lattice; this application is particularly intriguing since the lattice structure defines special directions, which numerical simulations show are favoured by travelling waves. Also the motion of a wave through a hexagonal arrangement of atoms will differ from that through a square array of atoms. The project will involve a combination of theoretical and numerical techniques to the study such systems.