Spatial Resonance Overlap in Bose-Einstein Condensates in Superlattice Potentials

This paper, which I wrote jointly with one of my Georgia Tech undergraduate research students, was just published a couple days ago by International Journal of Bifurcation and Chaos. My former student is now a Ph.D. student in applied math at Stanford. (I converted her from a EE person to an applied mathematician. I am so proud of myself.)

The idea behind this paper is that a periodic lattice (represented by a sinusoid) is like a parametric excitation in space when one is looking at standing waves in Bose-Einstein condensates, which are describved by a cubic nonlinear Schrodinger equation. A superlattice (which I discussed briefly in earlier blog entries) consists of a sum of two sinusoids, so one can see a resonance with respect to either of the two forcings. (By the way, the canonical physical example of a parametric excitation is a vertically vibrated pendulum. A somewhat more complciated situation is the Faraday system, in which a tub of water is oscillated vertically. Mathematically, the simplest equation describing a parametric excitation is the Mathieu equation.) Each of the forcings can separately cause some chaotic behavior, which increases in significance (more precisely, in the area of the phase space that is chaotic) as the forcing amplitude is increased. When more than one forcing is present, at some point the resonance zones from the separate forcings start to overlap and this leads to the onset of globally chaotic dynamics. How strong a forcing is required for this to happen can be estimated crudely using what is known as "Chirikov's overlap criterion." In our paper, my student (Vivien Chua) and I did this. Naturally, as this was a student project, she did nearly all of the work. I was the advisor.