This study aims at presenting a grid for analysing the way the language employed in Greek school science textbooks tends to project pedagogic messages. These messages are analysed for the different school science subjects (i.e., Physics, Chemistry, Biology) and educational levels (i.e., primary and lower secondary level). The analysis is made using the dimensions of content specialisation (classification) and social-pedagogic relationships (framing) promoted by the language of the school science textbooks as well as the elaboration and abstraction of the corresponding linguistic code (formality), thus combining pedagogical and socio-linguistic perspectives. Classification and formality are used to identify the ways science textbooks tend to position students in relation to the interior of the corresponding specialised body of knowledge (i.e., in terms of content and code) while framing is used to identify the ways science textbooks tend to position students as learning subjects within the school science discourse. The results show that the kind of pedagogic messages projected by the textbooks depends mainly on the educational level and not particularly on the specific discipline. As the educational level rises a gradual move towards more specialised forms of scientific knowledge (mainly in terms of code) with a parallel increase in the students' autonomy in accessing the textbook material is noticed. The implications concern the way both students and teachers approach science textbooks as well as the roles they can undertake by internalising the textbooks' pedagogic messages and also the way science textbooks are authored.

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We consider space discretizations of the matrix Zakharov-Shabat AKNS scheme, in particular the discrete matrix non-linear Scrhrödinger (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. 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.

We focus on the non-linear Schrödinger 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.

We consider the generalized matrix non-linear Schrödinger (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 matrixdifferential 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.