• Addendum to the article: On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the Half-Line

      Antonopoulou, Dimitra; Kamvissis, Spyridon (IOPSCIENCE Published jointly with the London Mathematical Society, 2016-08-31)
      We present a short note on the extension of the results of [1] to the case of non-zero initial data. More specifically, the defocusing cubic NLS equation is considered on the half-line with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable.
    • On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the half-line

      Antonopoulou, Dimitra; Kamvissis, Spyridon; Department of Mathematics, University of Chester, UK(D.A) and Department of Mathematics and Applied Mathematics, University of Crete, Greece (S.K) (IOPSCIENCE Published jointly with the London Mathematical Society, 2015-07-24)
      Initial-boundary value problems for 1-dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a well-posed problem. In the case of cubic NLS, knowledge of the Dirichet data su ces to make the problem well-posed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also su ciently decaying and that, hence, the Fokas method can be applied.
    • On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system

      Kavallaris, Nikos I.; Suzuki, Takashi; University of Chester; Osaka University (London Mathematical Society, 2017-03-21)
      The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.