Modulated amplitude waves in collisionally inhomogeneous Bose–Einstein condensates

We investigate the dynamics of an effectively one-dimensional Bose–Einstein condensate (BEC) with scattering length aa subjected to a spatially periodic modulation, a=a(x)=a(x+L)a=a(x)=a(x+L). This “collisionally inhomogeneous” BEC is described by a Gross–Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of xx. We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that “on-site” solutions, whose maxima correspond to maxima of a(x)a(x), are more robust and likely to be observed than their “off-site” counterparts.