A Galërkin projection and multiple scales approach to Feshbach resonance in Bose-Einstein condensates

In this paper, solutions of the nonlinear Schrödinger equation with a parabolic and a periodic potential modelling the dynamics of Bose-Einstein condensates are considered. A Galërkin projection approach is applied to reduce the partial differential equation to a system of nonlinear ordinary differential equations. In the case of Feshbach resonance, a multiple scales approach is applied to the reduced equations and is used to capture the dynamics of the full behavior of the PDE. Averaging is also used when the period of the oscillations corresponds to that of the difference in eigenvalues, thus inducing a resonance in the reduced equations. Accurate predictions of the overall behavior of the PDE can be made through this simplified model, including some very interesting resonance results.