Barbashin conditions for uniform instability of evolution operators

"The aim of the present paper is to give some characterization theorems of Barbashin type for the uniform exponential instability and uniform polynomial instability behavior of evolution operators. Also, some examples which illustrate the connections between the concepts presented are given."

Stability of single and multiple matter-wave dark solitons in collisionally inhomogeneous Bose–Einstein condensates

We examine the spectral properties of single and multiple matter-wave dark solitons in Bose–Einstein condensates confined in parabolic traps, where the scattering length is periodically modulated. In addition to the large density limit picture previously established for homogeneous nonlinearities, we explore a perturbative analysis in the vicinity of the linear limit, which provides good agreement with the observed spectral modes. Between these two analytically tractable limits, we use numerical computations to fill in the relevant intermediate regime. We find that the scattering length modulation can cause a variety of features absent for homogeneous nonlinearities. Among them, we note the potential oscillatory instability even of the single dark soliton, the potential absence of instabilities in the immediate vicinity of the linear limit for two dark solitons, and the existence of an exponential instability associated with the in-phase motion of three dark solitons.

On nonuniform exponential stability for skew-evolution semiflows in Banach spaces

The paper considers some concepts of nonuniform asymptotic stability for skew-evolution semiflows in Banach spaces, which we have introduced in [Megan, M. and Stoica, C., Exponential instability of skew-evolution semiflows in Banach spaces, Stud. Univ. Babes-Bolyai Math., LIII (2008), No. 1, 17–24] and for which we present equivalent definitions, as well as integral characterizations in a nonuniform setting. Some examples are included to illustrate the results and to clarify the differences between the uniform and nonuniform cases.

Polynomial instability of evolution operators in Banach spaces

In this paper we consider two concepts of polynomial instability for evolution operators in Banach spaces. Our approach is based on the extension of techniques for exponential instability to the case of polynomial instability. The obtained results are generalizations of some theorems about uniform and nonuniform exponential instability.