Singular instability of exact stationary solutions of the nonlocal Gross-Pitaevskii equation

TL;DR

This paper demonstrates that nonlocal perturbations in nonlinear Schrödinger systems can cause instabilities in solutions that are stable in the local case, with a focus on the nonlocal Gross-Pitaevskii equation.

Contribution

It constructs exact stationary solutions for the nonlocal Gross-Pitaevskii equation and analyzes how nonlocality induces singular instability as it approaches the local limit.

Findings

Nonlocality causes beyond-all-orders instability in stable local solutions.

The instability onset is independent of the interaction kernel form.

Exact solutions are constructed for arbitrary interaction kernels.

Abstract

In this paper we show numerically that for nonlinear Schrodinger type systems the presence of nonlocal perturbations can lead to a beyond-all-orders instability of stable solutions of the local equation. For the specific case of the nonlocal one-dimensional Gross-Pitaevskii equation with an external standing light wave potential, we construct exact stationary solutions for an arbitrary interaction kernel. As the nonlocal and local equations approach each other (by letting an appropriate small parameter $\epsilon\to 0$), we compare the dynamics of the respective solutions. By considering the time of onset of instability, the singular nature of the inclusion of nonlocality is demonstrated, independent of the form of the interaction kernel.