A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov--Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data

Monosynaptic pathway from rat vibrissa motor cortex to facial motor neurons revealed by lentivirus-based axonal tracing

The mammalian motor cortex typically innervates motor neurons indirectly via oligosynaptic pathways. However, evolution of skilled digit movements in humans, apes, and some monkey species is associated with the emergence of abundant monosynaptic cortical projections onto spinal motor neurons innervating distal limb muscles. Rats perform skilled movements with their whiskers, and we examined the possibility that the rat vibrissa motor cortex (VMC) projects monosynaptically onto facial motor neurons controlling the whisker movements. First, single injections of lentiviruses to VMC sites identified by intracortical microstimulations were used to label a distinct subpopulation of VMC axons or presynaptic terminals by expression of enhanced green fluorescent protein (GFP) or GFP-tagged synaptophysin, respectively. Four weeks after the injections, GFP and synaptophysin-GFP labeling of axons and putative presynaptic terminals was detected in the lateral portion of the facial nucleus (FN), in close proximity to motor neurons identified morphologically and by axonal back-labeling from the whisker follicles. The VMC projections were detected bilaterally, with threefold larger density of labeling in the contralateral FN. Next, multiple VMC injections were used to label a large portion of VMC axons, resulting in overall denser but still laterally restricted FN labeling. Ultrastructural analysis of the GFP-labeled VMC axons confirmed the existence of synaptic contacts onto dendrites and somata of FN motor neurons. These findings provide anatomical demonstration of monosynaptic VMC-to-FN pathway in the rat and show that lentivirus-based expression of GFP and GFP-tagged presynaptic proteins can be used as a high-resolution neuroanatomical tracing method

Синтез та біологічна активність трифторометилзаміщених анілідів 4-гідрокси-2,2-діоксо-1Н-2λ6,1-бензотіазин-3-карбонових кислот

In order to reveal the regularities of the “structure – biological activity” relationship by interaction of esters of 1-R-4-hydroxy-2,2-dioxo-1H-2λ6,1-benzothiazin-3-carboxylic acids and trifluoromethyl substituted anilines in boiling xylene with good yields and purity the corresponding N-aryl-4-hydroxy-2,2-dioxo-1H-2λ6,1-benzothiazin-3-carboxamides have been synthesized. The structure of the compounds obtained has been confirmed by the data of elemental analysis and NMR 1Hspectroscopy. It has been shown that the presence of trifluoromethyl groups having the powerful electron-withdrawing properties affects the position of signals of the aniline moiety protons: comparing to the spectra of the model methyl derivatives they undergo a significant paramagnetic shift. According to the results of the pharmacological studies conducted it has been found that the replacement of methyl groups in the anilide moiety of 1-R-4-hydroxy-2,2-dioxo-1H-2λ6,1-benzothiazin-3-carboxamides to trifluoromethyl has a different effect on their analgesic activity, which can remain at the original level, be completely lost or significantly increase. However, N-aryl-4-hydroxy-2,2-dioxo-1H-2λ6,1-benzothiazin-3-carboxamides definitely lose the ability to influence in any way on the excretory renal function after this chemical modification.С целью выявления закономерностей связи «структура – биологическая активность» взаимодействием сложных эфиров 1-R-4-гидрокси-2,2-диоксо-1H-2λ6,1-бензотиазин-3-карбоновых кислот и трифторметилзамещенных анилинов в кипящем ксилоле с хорошими выходами и чистотой синтезированы соответствующие N-арил-4-гидрокси-2,2-диоксо-1Н-2λ6,1-бензотиазин-3-карбоксамиды. Строение полученных соединений подтверждено данными элементного анализа и спектроскопии ЯМР 1Н. Показано, что присутствие обладающих мощными электроноакцепторными свойствами трифторметильных групп сказывается на положении сигналов протонов анилидных фрагментов – по сравнению со спектрами модельных метильных производных они претерпевают существенный парамагнитный сдвиг. По результатам проведенных фармакологических испытаний найдено, что замена метильных групп в анилидном фрагменте 1-R-4-гидрокси-2,2-диоксо-1Н-2λ6,1-бензотиазин-3-карбоксамидов на трифтор-метильные по-разному влияет на их анальгетическую активность, которая может оставаться на исходном уровне, полностью теряться или значительно усиливаться. А вот способность влиять каким-либо образом на мочевыделительную функцию почек N-арил-4-гидрокси-2,2-диоксо-1Н-2λ6,1-бензотиазин-3-карбоксамиды после указанной химической модификации однозначно утрачивают.З метою виявлення закономірностей зв’язку «структура – біологічна активність» взаємодією естерів 1-R-4-гідрокси-2,2-діоксо-1H-2λ6,1-бензотіазин-3-карбонових кислот та трифторометилзаміщених анілінів у киплячому ксилолі з добрими виходами і чистотою синтезовані відповідні N-арил-4-гідрокси- 2,2-діоксо-1Н-2λ6,1-бензотіазин-3-карбоксаміди. Будова одержаних сполук доведена даними елементного аналізу та спектроскопії ЯМР 1Н. Показано, що присутність трифторометильних груп, які виявляють сильні електроноакцепторні властивості, позначається на положенні сигналів протонів анілідних фрагментів – порівняно зі спектрами модельних метильних похідних вони піддаються суттєвому парамагнітному зсуву. За результатам проведених фармакологічних випробовувань знайдено, що заміна метильних груп в анілідному фрагменті 1-R-4-гідрокси-2,2-діоксо-1Н-2λ6,1-бензотіазин-3-карбоксамідів на трифторометильні по-різному впливає на їх аналгетичну активність, яка може залишатися на вихідному рівні, повністю втрачатися або ж значно посилюватися. А ось здатність впливати будь-яким чином на сечовидільну функцію нирок N-арил-4-гідрокси-2,2-діоксо-1Н-2λ6,1-бензотіазин-3-карбоксаміди після зазначеної хімічної модифікації однозначно втрачають

The Gould-Hopper Polynomials in the Novikov-Veselov equation

We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the $\sigma$-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particulr, when $N=3,4,5$, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.Comment: 22 pages, no figur

Towards an Inverse Scattering theory for non decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe $N$ solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a DBAR-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N=1 and N=2 are investigated in details. The singularities of the resolvent for the case N=1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.Comment: 22 pages, submitted to Inverse Problem

Topological Phenomena in the Real Periodic Sine-Gordon Theory

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure

Isoperiodic deformations of the acoustic operator and periodic solutions of the Harry Dym equation

We consider the problem of describing the possible spectra of an acoustic operator with a periodic finite-gap density. We construct flows on the moduli space of algebraic Riemann surfaces that preserve the periods of the corresponding operator. By a suitable extension of the phase space, these equations can be written with quadratic irrationalities.Comment: 15 page