Unstable gap solitons in inhomogeneous Schrodinger equations

TL;DR

This paper investigates the existence, construction, and stability of gap solitons in inhomogeneous Schrödinger equations with periodic coefficients, including analytical methods and numerical validation.

Contribution

It introduces a phase-space and topological approach to construct and analyze the stability of gap solitons in inhomogeneous media.

Findings

Existence of gap solitons in finite spectral band gaps.

Construction of gap solitons via phase portrait methods.

Numerical validation of analytical stability results.

Abstract

A periodically inhomogeneous Schrodinger 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, characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, how to construct such gap solitons through a composite phase portrait is shown. By exploiting the phase-space method and combining it with the application of a topological argument, it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Surface gap solitons at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed. Numerical calculations are presented accompanying the analytical results.