Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations and initial conservation laws, follow from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for generation of symmetries is developed and applied to discrete KdV. It can also be applied to all the other ABS equations

Infinitely many conservation laws for the discrete KdV equation

In \cite{RH3} Rasin and Hydon suggested a way to construct an infinite number of conservation laws for the discrete KdV equation (dKdV), by repeated application of a certain symmetry to a known conservation law. It was not decided, however, whether the resulting conservation laws were distinct and nontrivial. In this paper we obtain the following results: (1) We give an alternative method to construct an infinite number of conservation laws using a discrete version of the Gardner transformation. (2) We give a direct proof that the Rasin-Hydon conservation laws are indeed distinct and nontrivial. (3) We consider a continuum limit in which the dKdV equation becomes a first-order eikonal equation. In this limit the two sets of conservation laws become the same, and are evidently distinct and nontrivial. This proves the nontriviality of the conservation laws constructed by the Gardner method, and gives an alternate proof of the nontriviality of the conservation laws constructed by the Rasin-Hydon method

Construction of Integrals of Higher-Order Mappings

We find that certain higher-order mappings arise as reductions of the integrable discrete A-type KP (AKP) and B-type KP (BKP) equations. We find conservation laws for the AKP and BKP equations, then we use these conservation laws to derive integrals of the associated reduced maps.Comment: appear to Journal of the Physical Society of Japa

An algebraic method of classification of S-integrable discrete models

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.Comment: 11 pages, workshop "Nonlinear Physics. Theory and Experiment VI", submitted to TM

The lattice Schwarzian KdV equation and its symmetries

In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized symmetries and master symmetries. We finally show that we can use master symmetries of the lSKdV equation to construct non-autonomous non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE VI

Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of isospectral and non-isospectral zero curvature representations are derived for all of them.Comment: 22 page

Continuous symmetric reductions of the Adler-Bobenko-Suris equations

Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi "generating partial differential equations" is established and an auto-Backlund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1$_{\delta=0}$ members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents.Comment: 28 pages, submitted to J. Phys. A: Math. Theo

Classification of integrable discrete Klein-Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models. A new integrable example is found, its higher symmetry is presented.Comment: 12 pages, submitted to Physica Script

Experience and features at studying of bases of gerontology at students of stomatologic faculty in introduction in credit-module system

У статті викладені аспекти викладання основ геронтології та геріатрії в клініці внутрішніх хвороб, яке проводитиметься на кафедрі внутрішньої медицини з метою вдосконалення знань і практичних навичок у студентів 3 курсу, які навчаються за спеціальністю «стоматологія» в умовах запровадження кредитно-модульної системи з використанням системи програмірованной контрольованої самостійної роботи студентів. Основною метою курсу є вивчення студентами вікових змін, які відбуваються в старіючому організмі, особливостей перебігу захворювань у осіб похилого та старечого віку, методів профілактики, основних принципів фармакотерапії

Профілізація - основа підвищення якості підготовки cтоматологів із внутрішніх хвороб

Наведено досвід роботи кафедри внутрішньої медицини № 3 ВДНЗУ «УМСА». Показано, що протягом останніх 20 років пріоритетним завданням кафедри було максимальне наближення викладання внутрішніх хвороб до реальних потреб лікаря-стоматолога. Це стосується впровадження нових програм із дисципліни (остання в редакції 2015 року), нових підручників і навчальних посібників, зближення змісту і методики навчання студентів стоматологічного факультету зі світовою та європейською практикою