On the classification of scalar evolutionary integrable equations in 2+12+1 dimensions

We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper \cite{FMN} provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifth order analogues and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality $w$. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic reductions of the dispersionless limit by the full dispersive equation

Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies

Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems. Different levels of generalized hierarchy are connected via invariants of Combescure symmetry transformation. Resolution of functional equations also leads to the $\tau$-function and addition formulae to it.Comment: 43 pages, Late

Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations

We obtain new gauge-invariant forms of two-dimensional integrable systems of nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov system. We show how these forms imply both new and well-known two-dimensional integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and modified Nizhnik-Veselov-Novikov equation. We consider Miura-type transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment. V", Gallipoli (Lecce, Italy), 12-21 June, 200

Gauge-invariant description of some (2+1)-dimensional integrable nonlinear evolution equations

New manifestly gauge-invariant forms of two-dimensional generalized dispersive long-wave and Nizhnik-Veselov-Novikov systems of integrable nonlinear equations are presented. It is shown how in different gauges from such forms famous two-dimensional generalization of dispersive long-wave system of equations, Nizhnik-Veselov-Novikov and modified Nizhnik-Veselov-Novikov equations and other known and new integrable nonlinear equations arise. Miura-type transformations between nonlinear equations in different gauges are considered.Comment: 13 pages, LaTeX, no figure

Recursion operator for stationary Nizhnik--Veselov--Novikov equation

We present a new general construction of recursion operator from zero curvature representation. Using it, we find a recursion operator for the stationary Nizhnik--Veselov--Novikov equation and present a few low order symmetries generated with the help of this operator.Comment: 6 pages, LaTeX 2

Relation between hyperbolic Nizhnik-Novikov-Veselov equation and stationary Davey-Stewartson II equation

A Lax system in three variables is presented, two equations of which form the Lax pair of the stationary Davey-Stewartson II equation. With certain nonlinear constraints, the full integrability condition of this Lax system contains the hyperbolic Nizhnik-Novikov-Veselov equation and its standard Lax pair. The Darboux transformation for the Davey-Stewartson II equation is used to solve the hyperbolic Nizhnik-Novikov-Veselov equation. Using Darboux transformation, global $n$-soliton solutions are obtained. It is proved that each $n$-soliton solution approaches zero uniformly and exponentially at spatial infinity and is asymptotic to $n^2$ lumps of peaks at temporal infinity.Comment: 25 pages, 5 figure

Vitamin D3 (Cholecalciferol) and Pelvic Pain Caused by Ovarian Endometriosis

Статья посвящена изучению взаимосвязи концентрации витамина D с наличием и интенсивностью тазовой боли у женщин, страдающих эндометриозом яичников. Цель исследования: установить взаимосвязь уровня витамина D и интенсивности тазовой боли у женщин, страдающих эндометриозом яичников. В исследование были включены 190 женщин репродуктивного возраста с верифицированным диагнозом эндометриоз яичников, в возрасте от 20 до 41 года (средний возраст – 29,1±3,3 года). Для определения интенсивности тазовой боли использовали визуальную аналоговую шкалу (ВАШ). Всем пациенткам (n=190) исходно оценку уровня витамина D проводили путем определения уровня общего 25(ОН)D в сыворотке крови методом масс-спектрометрии, адаптированным к клинической практике, согласно международным стандартам, на тандемном масс-спектрометре AB SCIEX QTRAP 5500.The article studies the relation between vitamin D concentration and the presence and intensity of pelvic pain in women suffering from ovarian endometriosis. The aim of the study is to examine the relation between vitamin D levels and intensity of pelvic pain in women with ovarian endometriosis. Study design: cohort, prospective, non-comparative study. The study included 190 women of reproductive age suffering from ovarian endometriosis (OE), aged 20 to 41 years (mean age 29.1±3.3 years). Visual analogue scale (VAS) was used to determine the intensity of pelvic pain.The evaluation of vitamin D level in all patients (n=190) was initially performed by determining the level of total 25 (OH) D in blood serum by the method of mass spectrometry. Study Results: The average vitamin D level in the study was 23.98±6.82 ng/ml. The average concentration of vitamin D in the blood of the patients with low-intensity pelvic pain was 26.7±5.92 ng/ml, with moderate-intensity – 23.06±of 5.55 ng/ml, with high pain intensity – 19.26±6,01 ng/ml. The content of vitamin D in the blood of the patients in the control group was 28.83±6.15 ng/ml, which met the criteria of insufficiency. Аn inverse correlation between the severity of pain based on VAS and vitamin D in women with endometriosis of the main group was established in the study, it amounted to 0.502 (p<0.001)

Полтавський національний технічний університет імені Юрія Кондратюка

В статье рассмотрены отдельные элементы сталежелезобетонные безбалочными перекрытия - сборные надколонни плиты перекрытия со стальным обрамлением. Особенностью предлагаемой конструкции является то, что стальные уголки обрамления работают совместно с железобетонной составляющей, при этом они применяются в качестве несъемной опалубки. Приведенные результаты экспериментальных исследований и их анализ.У статті розглянуті окремі елементи сталезалізобетонного безбалкового перекриття – збірні надколонні плити перекриття зі сталевим обрамленням. Особливістю запропонованої конструкції є те, що сталеві кутики обрамлення працюють сумісно із залізобетонною складовою, при цьому вони застосовуються в якості незнімної опалубки. Наведені результати експериментальних досліджень та їх аналіз