An Instability Criterion for Nonlinear Standing Waves on Nonzero Backgrounds

In this work, we study the standing wave solutions of an inhomogeneous nonlinear Schrödinger equation. The inhomogeneity considered here is a varying coefficient of the nonlinear term. In particular, the nonlinearity is chosen to be repelling (defocusing) except on a finite interval. Localized solutions on a nonzero background, e.g., dark solitons trapped by the inhomogeneity, are identified and studied. A novel instability criterion for such states is established through a topological argument. This allows instability to be determined quickly in many cases by considering simple geometric properties of the standing waves as viewed in the composite phase plane. Numerical calculations accompany the analytical results.