The governing equations for the wave function of a Bose-Einstein condensate (BEC) are Hamiltonian; in particular, they are a variant of the nonlinear Schrodinger equation known as the Gross-Pitaevskii equation. The analysis of this physical system leads to several intriguing fundamental mathematical problems for Hamiltonian systems which deal with the existence, spectral stability, and nonlinear stability of waves. The major mathematical problems to be addressed in this proposal include (a) the construction and spectral stability of fully three-dimensional waves in the presence of a cigar magnetic trap via the technique of spatial dynamics, (b) the construction and spectral stability of two-dimensional periodic waves in the presence of an optical lattice, and (c) a careful study of eigenvalues with negative Krein signature. Regarding (c), there are two avenues that are to be explored: (1) the determination of embedded eigenvalues of negative sign in the case that the solitary wave is realized as a limit of a family of periodic waves, and (2) the determination of the exact number of eigenvalues with negative sign.
Especially since the 1997 Nobel Prize winning work of S. Chu, C. Cohen-Tannoudji, and W. Phillips on the exotic quantum phenomenon known as BECs, there has been a great deal of exciting experimental and theoretical work in the study of BECs. In order to form a BEC, it is necessary that the matter is cooled to billionths of a degree above absolute zero; hence, BECs are extremely fragile, and it is likely to be some time before any practical applications are developed. Nevertheless, they have proved to be useful in exploring a wide range of questions in fundamental physics. Examples include experiments that have demonstrated interference between condensates due to wave-particle duality, the study of superfluidity and quantized vortices, and the slowing of light pulses to very low speeds. Vortices in BECs are also currently the subject of analogue-gravity research, studying the possibility of modeling black holes and their related phenomena in the lab. The mathematical results of this proposal will help both theoreticians and experimentalists better understand the dynamics of not only patterns such as vortices, necklaces, and solitons in BECs, but also the dynamics of waves and patterns in Hamiltonian systems which are used to model interesting phenomena such as waves in fluids and light propagation in optical fibers. The inclusion and training of undergraduate students is an integral part of this proposal. Introducing students to the interplay between applications and experiments, numerics, and formal and rigorous analysis, will lead them to being excited about seeing the connections between the physical world and the mathematical world. This will further lead the participating students towards a deeper appreciation for the usefulness of mathematics in other disciplines, and perhaps then allow them to be open to the idea of doing more serious applied mathematics at the graduate level. Calvin College has a history of producing successful Ph.D. students in mathematics and statistics. Approximately one-third of these students have been women, who are significantly underrepresented in the mathematical sciences. Furthermore, Calvin has been very successful in the education and training of secondary education teachers. As a consequence of this award, the investigator will be able to fruitfully support the college-wide effort in training future researchers and educators in the mathematical sciences.
