Matter-wave solitons with a periodic, piecewise-constant scattering length

Physical Review A just published one of my articles: Matter-wave solitons with a periodic, piecewise-constant scattering length.

My coauthors are A. S. Rodrigues, P. G. Kevrekidis, D. J. Frantzeskakis [who I met in real life for the first time at the conference I'm currently attending] , P. Schmelcher, and A. R. Bishop. [I have yet to meet Augusto Rodrigues, Peter Smelcher, and Alan Bishop in person.]

Here is our abstract: Motivated by recent proposals of “collisionally inhomogeneous” Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we study the existence and stability properties of bright and dark matter-wave solitons of a BEC characterized by a periodic, piecewise-constant scattering length. We use a "stitching" approach to analytically approximate the pertinent solutions of the underlying nonlinear Schrödinger equation by matching the wave function and its derivatives at the interfaces of the nonlinearity coefficient. To accurately quantify the stability of bright and dark solitons, we adapt general tools from the theory of perturbed Hamiltonian systems. We show that stationary solitons must be centered in one of the constant regions of the piecewise-constant nonlinearity. We find both stable and unstable configurations for bright solitons and show that all dark solitons are unstable, with different instability mechanisms that depend on the soliton location. We corroborate our analytical results with numerical computations.


The idea behind this is that my collaborators and I (as well as others) have done some work on BECs in "nonlinear lattices" in which the nonlinearity coefficient (which is proportional to the two-body scattering length) is a periodic function of space: g = g(x). In my past paper on this topic, my collaborators and I looked at periodic waves and g(x) given by a trig function. To try to delve more deeper into some things analytically, we decided to take a step back and let g(x) be a periodic step function. One can solve the governing partial differential equation in closed form in the g(x) = constant regions and then one can try to match the solutions at the boundaries between those regions. This was motivated by some conversations with more theoretical mathematicians (Bjorn Sandstede and Percy Deift) and to try to find a setup that would be more tractable to some theorem-proof work by people closer to the pure side of the mathematical spectrum. In the just-published paper, we looked at localized solutions. Our plan is to look at periodic solutions (in the form of elliptic functions) as well.