Singular instability of exact stationary solutions of the non-local Gross-Pitaevskii equation

In this Letter we show numerically that for non-linear Schrödinger type equations the presence of non-local perturbations can lead to a singular instability of stable solutions of the local equation. For the specific case of the non-local one-dimensional Gross-Pitaevskii equation with an external standing light wave potential, we construct exact stationary solutions for an arbitrary interaction kernel. As the non-local and local equations approach each other (by letting an appropriate small parameter ɛ→0), we compare the dynamics of the respective solutions. By considering the time of onset of instability, the singular nature of the inclusion of non-locality is demonstrated, independent of the form of the interaction kernel.