Unstable gap solitons in inhomogeneous Schrödinger equations

A periodically inhomogeneous Schrödinger equation is considered. The inhomogeneity is reflected through a non-uniform coefficient of the linear and non-linear term in the equation. Due to the periodic inhomogeneity of the linear term, the system may admit spectral bands. When the oscillation frequency of a localized solution resides in one of the finite band gaps, the solution is a gap soliton, which is characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, it is shown how to construct such gap solitons through a composite phase portrait. By exploiting the phase-space method and combining it with the application of a topological argument derived in [15], here it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Numerical calculations are presented accompanying the analytical results. In addition to gap solitons, surface gap solitons located at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed.