Artificial neural networks (ANNs) have been developed, implemented and tested on the basis of a four-year-long experimental data set, with the aim of analyzing the performance and clinical outcome of an existing medical ward, and predicting the effects that possible readjustments and/or interventions on the structure may produce on it. Advantages of the ANN technique over more traditional mathematical models are twofold: on one hand, this approach deals quite naturally with a large number of parameters/variables, and also allows to identify those variables which do not play a crucial role in the system dynamics; on the other hand, the implemented ANN can be more easily used by a staff of non-mathematicians in the unit, as an on-site predictive tool. As such, the ANN model is particularly suitable for the case study. The predictions from the ANN technique are then compared and contrasted with those obtained from a generalized kinetic approach previously proposed and tested by the authors. The comparison on the two case periods shows the ANN predictions to be somewhat closer to the experimental values. However, the mean deviations and the analysis of the statistical coefficients over a span of multiple years suggest the kinetic model to be more reliable in the long run, i.e., its predictions can be considered as acceptable even on periods that are quite far away from the two case periods over which the many parameters of the model had been optimized. The approach under study, referring to paradigms and methods of physical and mathematical models integrated with psychosocial sciences, has good chances of gaining the attention of the scientific community in both areas, and hence of eventually obtaining wider diffusion and generalization.

We derive novel dark-bright soliton solutions for the three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions. The solutions are obtained within the framework of a recently developed inverse scattering transform for the underlying nonlinear integrable partial differential equation, and unlike dark-bright solitons in the two component (Manakov) system in the same dispersion regime, their interactions display non-trivial polarization shift for the two bright components.

We use the Inverse Scattering Transform machinery to construct multi- soliton solutions to the 2-component defocusing nonlinear Schrodinger equation. Such solutions include dark–dark solitons, which have dark solitonic behaviour in both components, as well as dark–bright soliton solutions, with one dark and one bright component. We then derive the explicit expressions of two soliton solutions for all possible cases: two dark–dark solitons, two dark–bright solitons, and one dark–dark and one dark–bright soliton. Finally, we determine the long-time asymptotic behaviours of these solutions, which allows us to obtain explicit expressions for the shifts in the phases and in the soliton centers due to the interactions.

Soliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions are obtained which are the discrete analog of the Tajiri-Watanabe and Kutznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions are also presented. The latter, which can be thought of as a discrete “rogue” wave, is expressed as a family of rational functions of the discrete spatial variable n ∈ Z and time t ∈ R, parametrically depending on the amplitudeQo of the background. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus also providing a spectral characterization of the solutions and a description of the singular limit process.

We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schr¨odinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the magnitudes of the limiting values of the solution at space infinities are not the same). The theory is formulated without making use of Riemann surfaces, and by dealing explicitly instead with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit single-valued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root jump of the corresponding eigenvalues. We write the inverse problem as a discontinuous Riemann Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. We also give an expression for the trace formula and asymptotic phase difference. Finally, for comparison purposes, we also present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when different choices are made for the location of the branch cuts.

In this paper we study the propagation of optical pulses in an optical medium with coherent three level atomic transitions. The interaction between the pulses and the medium is described by the coupled Maxwell–Bloch equations, which we investigate by applying the method of inverse scattering transform. The details of the inverse scattering method and the non-trivial evolution of the associated scattering data are discussed. The one- and two-soliton solutions, polarization shifts due to two-soliton interactions, and the explicit form of the transmission matrix associated with pure soliton solutions are also derived.

The purpose of this paper is to construct the inverse scattering transform for the focusing Ablowitz-Ladik equation with nonzero boundary conditions at infinity. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem on a doubly connected curve in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential.

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrodinger (NLS) equation with one-sided non-zero boundary value as x → +∞ is presented. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that [q(x, t) − qr(t)ϑ(x)] ∈ L1,1(R) [here and in the following ϑ(x) denotes the Heaviside function] with respect to x ∈ R for all t ≥ 0, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations and as a Riemann-Hilbert problem on a single sheet of the scattering variables $λ_r = sqrt{k^2 + A^2_r}$, where k is the usual complex scattering parameter in the IST. The direct and inverse problems are also formulated in terms of a suitable uniformization variable that maps the two-sheeted Riemann surface for k into a single copy of the complex plane. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with the same amplitude as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the long-time asymptotic behavior of physically relevant NLS solutions with nontrivial boundary conditions, either via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.

The Inverse Scattering Transform (IST) for the defocusing vector nonlinear Schrodinger equations (NLS), with an arbitrary number of components and nonvanishing boundary conditions at space infinities, is formulated by adapting and generalizing the approach used by Beals, Deift, and Tomei in the development of the IST for the N-wave interaction equations. Specifically, a complete set of sectionally meromorphic eigenfunctions is obtained from a family of analytic forms that are constructed for this purpose. As in the scalar and two-component defocusing NLS, the direct and inverse problems are formulated on a two-sheeted, genus-zero Riemann surface, which is then transformed into the complex plane by means of an appropriate uniformization variable. The inverse problem is formulated as a matrix Riemann-Hilbert problem with prescribed poles, jumps, and symmetry conditions. In contrast to traditional formulations of the IST, the analytic forms and eigenfunctions are first defined for complex values of the scattering parameter, and extended to the continuous spectrum a posteriori.

A mathematical model, based on a statistical system approach, has been implemented and tested on the basis of a four-year-long experimental data set, with the aim of analyzing the performance and clinical outcome of an existing medical ward, and predicting the effects that possible readjustments and/or interventions on the structure may produce on it. The dynamics of the system is assumed to be connected to a variable called ‘‘atmosphere’’ that refers to the perceived social and organizational climate, as well as the comfort and ease realized in the ward. In this context, the atmosphere is intuitively related to the ‘‘quality’’ that is (or is perceived as being) offered by the service, as it affects the ability to satisfy the patients’ needs, to provide a livable environment for patients and medical staff, and to guarantee more efficient performances and a more complete professional development. Identifying variables, parameters and events that control the atmosphere is therefore of the deepest importance from a social and health-care point of view. The proposed interdisciplinary approach, referring to paradigms of physical and mathematical models integrated with theories and methods typical of social sciences, has chances of gaining the attention of the scientific community in both fields, and higher possibilities of obtaining appreciation and generalization.

We revisit the scattering problem for the defocusing nonlinear Schrodinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.

We study dark–bright soliton interactions in multi-component media such as nonlinear optical media in the defocusing regime and repulsive Bose–Einstein condensates. This is achieved using the recently developed formalism of the inverse scattering transform for the defocusing multi-component nonlinear Schrödinger equation with non-zero boundary conditions. We show that, generically, these interactions result in a non-trivial polarization shift for the bright components. We compute such polarization shift analytically and compare it to that in focusing two-component nonlinear Schrödinger systems.

A rigorous theory of the inverse scattering transform for the defocusing nonlinear Schrödinger equation with nonvanishing boundary values is presented. The direct problem is shown to be well posed for potentials in a suitable functional class, for which analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated and solved both via Marchenko integral equations, and as a Riemann-Hilbert problem in terms of a suitable uniform variable. The asymptotic behavior of the scattering data is determined and shown to ensure the linear system solving the inverse problem is well defined. Finally, the triplet method is developed as a tool to obtain explicit multisoliton solutions by solving the Marchenko integral equation via separation of variables.

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schr¨odinger (NLS) equation with non-zero boundary values $q_{l/r} (t) ≡ A_{l/r} e−2i A^2_{l/r} t+iθ_{l/r} $as x →∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with $A_l ne A_r$ and $θ_l ne θ_r$ . The direct problem is shown to be well-defined for NLS solutions q(x, t) such that$q(x, t) − q_{l/r} (t)∈ L^{1,1}(R^{∓})$ with respect to x for all t ≥ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables $λ_{l/r} =sqrt{k^2 + A^2_{l/r}$ , where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x →±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the longtime asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as x approaches infinity. The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.