Classification of five-point differential-difference equations

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.Comment: 29 page

Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.Comment: 27 page

Emotional characteristics of sportsman students engaged in bullying and cyberbullying in relation to migrant students

The goal of this paper was to find a way to predict cyberbullying based on an assessment of the emotional insensitivity of university and college students. In addition, the goal was to study the intermediate mechanisms of regulating the relationship of emotional insensitivity and cyberbullying. The fundamental basis of this study was the assessment of emotional insensitivity using the method of "Determining the signs of emotional insensitivity". This assessment was made using a questionnaire that included three parts: callousness, indifference, and lack of emotion. This research has shown that empathy mitigates the adverse effects of emotional insensitivity on the processes of justification of immorality and according to the correlation analysis, there is a negative effect of the justification of immorality on cyberbullying. This study is an important step in revealing the emotional characteristics of students who are inclined to engage in cyberbullying. It is shown that the phenomenon of justification of cyberbullying is one of the potential mechanisms by which emotional features begin to affect the ability of students to engage in cyberbullying. Attention to the phenomenon of justification of cyberbullying reveals the relationship of emotional characteristics and cyberbullying. The conclusions of this study can be the basis for successful psychocorrection of students engaged in cyberbullying. The scientific novelty of this research is to reveal the mechanisms of interaction of emotional insensitivity, empathy and justification of immoral behaviour of students engaged in cyberbullying

Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation

We get the leading term of the Gurevich-Pitaevskii special solution to the KdV equation in the oscillation zone without using averaging methods.Comment: 13 pages, 3 figure

Evidence for Induced Magnetization in Superconductor-Ferromagnet Hetero-structures: a Scanning Tunnelling Spectroscopy Study

We performed scanning tunneling spectroscopy of c-axis oriented YBCO films on top of which ferromagnetic SRO islands were grown epitaxially in-situ. When measured on the ferromagnetic islands, the density of states exhibits small gap-like features consistent with the expected short range penetration of the order parameter into the ferromagnet. However, anomalous split-gap structures are measured on the superconductor in the vicinity of ferromagnetic islands. This observation may provide evidence for the recently predicted induced magnetization in the superconductor side of a superconductor/ ferromagnet junction. The length scale of the effect inside the superconductor was found to be an order of magnitude larger than the superconducting coherence length. This is inconsistent with the theoretical prediction of a penetration depth of only a few superconducting coherence lengths. We discuss a possible origin for this discrepancy

Classification of semidiscrete hyperbolic type equations. The case of third order symmetries

In this paper, a classification of semidiscrete equations of hyperbolic type is carried out. We study the class of equations of the form $$\frac{du_{n+1}}{dx}=f\left(\frac{du_{n}}{dx},u_{n+1},u_{n}\right),$$ here is the unknown function $u_n (x)$ depends on one discrete $n$ and one continuous $x$ variables. The classification is based on the requirement for the existence of higher symmetries in the discrete and continuous directions. The case is considered when the symmetries are of order 3 in both directions. As a result, a list of equations with the required conditions is obtained.Comment: 14 pages (in Russian