Time-like boundary conditions in the NLS model

We focus on the non-linear Schrodinger model and we extend the notion of space-time dualities in the presence of integrable time-like boundary conditions. We identify the associated time-like `conserved' quantities and Lax pairs as well as the corresponding boundary conditions. In particular, we derive the generating function of the space components of the Lax pairs in the case of time-like boundaries defined by solutions of the reflection equation. Analytical conditions on the boundary Lax pair lead to the time like-boundary conditions. The time-like dressing is also performed for the first time, as an effective means to produce the space components of the Lax pair of the associated hierarchy. This is particularly relevant in the absence of a classical r-matrix, or when considering complicated underlying algebraic structures. The associated time Riccati equations and hence the time-like conserved quantities are also derived. We use as the main paradigm for this purpose the matrix NLS-type hierarchy.Comment: 17 pages, LaTex. A few typos corrected. arXiv admin note: substantial text overlap with arXiv:1810.1093

Non-commutative NLS-type hierarchies: dressing & solutions

We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By employing the universal Darboux-dressing scheme we derive solutions for the hierarchy of integrable PDEs via solutions of the matrix Gelfand-Levitan-Marchenko equation, and we also identify recursion relations that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results are obtained considering either matrix-integral or general $n^{th}$ order matrix-differential operators as Darboux-dressing transformations. In this framework special links with the Airy and Burgers equations are also discussed. The matrix version of the Darboux transform is also examined leading to the non-commutative version of the Riccati equation. The non-commutative Riccati equation is solved and hence suitable conserved quantities are derived. In this context we also discuss the infinite dimensional case of the NLS matrix model as it provides a suitable candidate for a quantum version of the usual NLS model. Similarly, the non-commutitave Riccati equation for the general dressing transform is derived and it is naturally equivalent to the one emerging from the solution of the auxiliary linear problem.Comment: 29 pages, LaTex. Minor modification

Discretizations of the generalized AKNS scheme

We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhr\"odinger (DNLS) model, and the matrix generalization of the Ablowitz-Ladik (AL) model, which is the more widely acknowledged discretization. We focus on the derivation of solutions via local Darboux transforms for both discretizations, and we derive novel solutions via generic solutions of the associated discrete linear equations. The continuum analogue is also discussed, and as an example we identify solutions of the matrix NLS equation in terms of the heat kernel. In this frame we also derive a discretization of the Burgers equation via the analogue of the Cole-Hopf transform. Using the basic Darboux transforms for each scheme we identify both matrix DNLS-like and AL hierarchies, i.e. we extract the associated Lax pairs, via the dressing process. We also discuss the global Darboux transform, which is the discrete analogue of the integral transform, through the discrete Gelfand-Levitan-Marchenko (GLM) equation. The derivation of the discrete matrix GLM equation and associated solutions are also presented together with explicit linearizations. Particular emphasis is given in the discretization schemes, i.e. forward/backward in the discrete matrix DNLS scheme versus symmetric in the discrete matrix AL model.Comment: 28 pages, Latex. Typos corrected and clarifying comments added. Version accepted in J. Phys.

Towards an analysis of visual images in school science textbooks and press articles about science and technology

Abstract This paper aims at presenting the application of a grid for the analysis of the pedagogic functions of visual images included in school science textbooks and daily press articles about science and technology. The analysis is made using the dimensions of content specialisation (classification) and social-pedagogic relationships (framing) promoted by the images as well as the elaboration and abstraction of the corresponding visual code (formality), thus combining pedagogical and sociosemiotic perspectives. The grid is applied to the analysis of 2819 visual images collected from school science textbooks and another 1630 visual images additionally collected from the press. The results show that the science textbooks in comparison to the press material: a) use ten times more images, b) use more images so as to familiarise their readers with the specialised techno-scientific content and codes, and c) tend to create a sense of higher empowerment for their readers by using the visual mode. Furthermore, as the educational level of the school science textbooks (i.e., from primary to lower secondary level) rises, the content specialisation projected by the visual images and the elaboration and abstraction of the corresponding visual code also increases. The above results have implications for the terms and conditions for the effective exploitation of visual material as the educational level rises as well as for the effective incorporation of visual images from press material into science classes