The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces

The Hamiltonian representation for the hierarchy of Lax-type flows on a dual space to the Lie algebra of shift operators coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems is found by means of a specially constructed Backlund transformation. The Hamiltonian description for the corresponding set of squared eigenfunction symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable (2+1)-dimensional differential-difference systems and their triple Lax-type linearizations is analysed. The existence problem of a Hamiltonian representation for the coupled Lax-type hierarchy on a dual space to the central extension of the shift operator Lie algebra is solved also

Lax Integrable Supersymmetric Hierarchies on Extended Phase Spaces

We obtain via Bäcklund transformation the Hamiltonian representation for a Lax type nonlinear dynamical system hierarchy on a dual space to the Lie algebra of super-integral-differential operators of one anticommuting variable, extended by evolutions of the corresponding spectral problem eigenfunctions and adjoint eigenfunctions, as well as for the hierarchies of their additional symmetries. The relation of these hierarchies with the integrable by Lax (2|1+1)-dimensional supersymmetric Davey-Stewartson system is investigated

Вплив переробки неякісних ветпрепаратів та кормових добавок на стан довкілля

Considering the requirements of the European Union for the quality and safety of veterinary drugs, feed and fodder additives and contamination of the environment with waste production, topical issue is the waste of utilization veterinary products of not quality: veterinary drugs and feed additives. Conducted analysis of waste utilization methods of veterinary drugs, feed and fodder additives will help manufacturers of veterinary products to use such methods of destruction of defective raw materials and drugs, which increase the culture of production and promote the fight for the health of animals and poultry, and therefore are safe for human health, is practical and economically feasible, not causing the environmental damage, make it possible to destroy the waste to the extent of their education.Учитывая требования Европейского союза к качеству и безопасности ветеринарных препаратов, кормов, кормовых добавок и загрязнения внешней среды отходами производства, актуален вопрос утилизации отходов некачественной ветеринарной продукции: ветеринарных препаратов и кормовых добавок. Проведенный анализ методов утилизации отходов ветеринарных препаратов, кормов, кормовых добавок даст возможность производителям ветеринарной продукции использовать такие методы уничтожения некачественного сырья и препаратов, которые повышают культуру производства продукции и способствуют здоровью животных и птицы, практичны и экономически целесообразны, не приносят вреда внешней среде, дают возможность уничтожать отходы по мере их образования.Враховуючи вимоги Європейського союзу до якості та безпечності ветеринарних препаратів, кормів та кормових добавок і забруднення природного середовища відходами виробництва, актуальним є питання утилізації відходів неякісної ветеринарної продукції: ветеринарних препаратів та кормових добавок. Проведений аналіз методів утилізування відходів ветеринарних препаратів, кормів та кормових добавок допоможе виробникам ветеринарної продукції використовувати такі методи знищення неякісної сировини та препаратів, які підвищують культуру виробництва продукції та сприяють здоров’ю тварин і птиці, а відповідно є безпечними для здоров’я людей, є практичними та економічно доцільними, не наносять шкоди навколишньому середовищу, дають можливість знищувати відходи в міру їх утворення

Poisson structures on (non)associative noncommutative algebras and integrable Kontsevich type Hamiltonian systems

We have revisited the classical Poisson manifold approach, closely related to construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, we presented its natural and simple generalization allowing effectively to describe a wide class of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras

On the Complete Integrability of Nonlinear Dynamical Systems on Discrete Manifolds within the Gradient-Holonomic Approach

A gradient-holonomic approach for the Lax type integrability analysis of differentialdiscrete dynamical systems is devised. The asymptotical solutions to the related Lax equation are studied, the related gradient identity is stated. The integrability of a discrete nonlinear Schredinger type dynamical system is treated in detail.Comment: 20 page

The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation and Poisson structures constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed.Comment: 18 page

The generalized centrally extended Lie algebraic structures and related integrable heavenly type equations

There are studied Lie-algebraic structures of a wide class of heavenly type non-linear integrable equations, related with coadjoint flows on the adjoint space to a loop vector field Lie algebra on the torus. These flows are generated by the loop Lie algebras of vector fields on a torus and their coadjoint orbits and give rise to the compatible Lax-Sato type vector field relationships. The related infinite hierarchy of conservations laws is analysed and its analytical structure, connected with the Casimir invariants, is discussed. We present the typical examples of such equations and demonstrate in details their integrability within the scheme developed. As examples, we found and described new multidimensional generalizations of the Mikhalev-Pavlov and Alonso-Shabat type integrable dispersionless equation, whose seed elements possess a special factorized structure, allowing to extend them to the multidimensional case of arbitrary dimension