The aim of this proposal is to explore a number of novel, emerging directions in the context of atomic Bose-Einstein condensates (BECs). The project will extend the recent collaborative work of the PIs on the theme of vortex dynamics in trapped condensates. There, we will explore both connections with experimental results obtained by a collaborating experimental group at Amherst College and ones with other areas of mathematics and physics. These include most notably the spectral theory of such nonlinear coherent states, structural phase transitions thereof, reductions to particle-based dynamical models with interesting nonlinear bifurcation phenomena, and computational tools to monitor the existence, stability and dynamics of vortex clusters. Also, a multi-component generalization of these themes will be considered in the recently established direction of spin-orbit coupled BECs. These contain a more complex operator structure within the nonlinear problem, with a linear part featuring both a dispersive (Laplacian) and a Dirac-like part and the interplay of these terms and their impact on nonlinear states such as dark or bright solitons and vortices will be studied. The results in this direction will be compared to ongoing experiments by a collaborating experimental group at Washington State University.
This research is expected to provide a new generation of both theoretical and computational tools for studying nonlinear coherent structures with a particular view towards the pristine atomic physics setup of Bose-Einstein condensates. Furthermore, the theoretical/mathematical tools developed here will bear broader impacts towards areas such as spectral theory and nonlinear ordinary and partial differential equations, among others. It is also envisioned that our findings will create connections with a number of areas of Physics such as Fluid Dynamics, Nonlinear Optics and Statistical Mechanics (of Phase Transitions). On the other hand, the developed computational tools will explore the interface between numerical bifurcation theory, dynamical system and even Monte-Carlo/Molecular Dynamics techniques and their potential use for the physical system at hand. Importantly, the research will be a genuine synergy in truly Applied Mathematics, involving not only the development of theoretical methods and computational techniques, but also their direct connection with physical experiments. Finally, the project will be critically focused on sustaining a dynamic and multi-disciplinary team with a strong and diverse core of graduate students and hence will be consistently geared towards having a significant set of broader impacts. The relevant research will be disseminated via high quality journal publications both in Mathematics and in Physics and will be presented at Nonlinear Science and Mathematical Physics conferences both in the US and abroad.
