D.E. Pelinovsky and P.G. Kevrekidis

Periodic oscillations of dark solitons in parabolic potentials

We reformulate the Gross--Pitaevskii equation with an external parabolic potential as a discrete dynamical system, by using the basis of Hermite functions. We consider small amplitude stationary solutions with a single node, called dark solitons, and examine their existence and linear stability. Furthermore, we prove the persistence of a periodic motion in a neighborhood of such solutions. Our results are corroborated by numerical computations elucidating the existence, linear stability and dynamics of the relevant solutions.

Gross-Pitaevskii equation, dark solitons, Hermite functions, discrete dynamical systems, existence and stability, Lyapunov theorem on periodic orbits