Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schr\"odinger Equation

We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use a finite-gap potential of the Dirac operator given in an explicit form through hyper-elliptic theta-functions. The paper aim is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse scattering technics. We investigate this problem exceptionally using the Riemann-Hilbert problems technics that allow us to obtain explicit formulas for the asymptotic solitons themselves that in contrast with the cited papers where asymptotic formulas are obtained only for the square of absolute value of solution. Using transformations of the main RH problems we arrive to a model problem corresponding to the parametrix at the end points of continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials which are naturally appeared after appropriate scaling of the Riemann-Hilbert problem in a small neighborhoods of the end points of continuous spectrum. Further asymptotic analysis give an explicit formula for solitons at the edge of dispersive wave. Thus, we give the complete description of the train of asymptotic solitons: not only bearing envelope of each asymptotic soliton, but its oscillating structure are found explicitly. Besides the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock wave.Comment: 36 pages, 5 figure

INTRA-OCEANIC ARCS OF THE PALEO-ASIAN OCEAN

Intra-oceanic arcs (IOAs) form at Pacific-type convergent margins, in the upper “stable” plate, when the subducting plate submerges to the depths of melting, i.e., to ca. 50–100 km. A typical IOA system, such as Mariana-Bonin and the Philippines Sea, consists of subduction zone, fore-arc region with accretionary prism, frontal or active arc, marginal basin with spreading center, and, in some cases, one or more remnant arcs and inactive marginal basin. The IOAs are very important elements of Pacific-type convergent margins as they represent major sites of juvenile continental crust formation (e.g. [Clift et al., 2003; Stern, 2010; Maruyama et al., 2011]), but are also the most important sites of crust removal by sediment subduction and tectonic/ subduction erosion [Stern, Scholl, 2010].Intra-oceanic arcs (IOAs) form at Pacific-type convergent margins, in the upper “stable” plate, when the subducting plate submerges to the depths of melting, i.e., to ca. 50–100 km. A typical IOA system, such as Mariana-Bonin and the Philippines Sea, consists of subduction zone, fore-arc region with accretionary prism, frontal or active arc, marginal basin with spreading center, and, in some cases, one or more remnant arcs and inactive marginal basin. The IOAs are very important elements of Pacific-type convergent margins as they represent major sites of juvenile continental crust formation (e.g. [Clift et al., 2003; Stern, 2010; Maruyama et al., 2011]), but are also the most important sites of crust removal by sediment subduction and tectonic/ subduction erosion [Stern, Scholl, 2010]

How to superize Liouville equation

So far, there are described in the literature two ways to superize the Liouville equation: for a scalar field (for $N\leq 4$) and for a vector-valued field (analogs of the Leznov--Saveliev equations) for N=1. Both superizations are performed with the help of Neveu--Schwarz superalgebra. We consider another version of these superLiouville equations based on the Ramond superalgebra, their explicit solutions are given by Ivanov--Krivonos' scheme. Open problems are offered

Step-Initial Function to the MKdV Equation: Hyper-Elliptic Long-Time Asymptotics of the Solution

The modified Korteveg{de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. q(x, 0) = cr for x ≥ 0 and q(x, 0) = cl for x cr > 0. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t → ∞.Рассматривается модифицированное уравнение КдФ на всей прямой с начальным условием типа ступеньки, которая равна константе cl при x 0. При этом выполняется условие cl > cr > 0, что обеспечивает режим "гидродинамической волны сжатия" при t → ∞. Цель статьи - изучение асимптотического поведения решения начально-краевой задачи, когда t → ∞

Electrophysiological Parameters of Sinus Node Function in Patients with Paroxysmal Tachyarrhythmias

to analyze the indicators of the function of the sinus node in patients of young age with paroxysmal tachycardia. Methods: study included 11 patients with suspected paroxysmal tachycardia, with an average age of 17±28. The basis for holding transesophageal electrophysiological study (TE EPS) was the clinical and electrophysiological characteristics of paroxysmal tachycardia. According to the results of Holter monitoring ECG (HM ECG) analyzed the minimum and maximum heart rate, number of ventricular and supraventricular arrhythmias, presence of pauses, rhythm and episodes of paroxysmal tachycardia. According to CHP, EFI estimated the initial heart rate (HR), recovery time of sinus node function (RTSNF), corrected recovery time of sinus node function (CRTSNF), point of Wenkebach (p. W), duration of the effective refractory period of the atrioventricular connections, presence of aberrant complexes and episodes of paroxysmal tachycardias before and after administration of atropine at a dose of 0.02 mg/kg. Results: Complaints characteristic of the tachyarrhythmia was diagnosed in 9 patients, episodes of heart rate more than 150 beats per minute in 7 patients. When conducting TE EPS obtained the following results: episodes of supraventricular tachycardia provoked in 8 patients (in two cases of paroxysmal tachycardia managed to provoke only after administration of atropine). Three of them have shimmer and atrial flutter episodes reciprocal tachycardia in five. Three patients provoke paroxysmal tachycardia failed, but they showed a shortening of the PQ interval and the appearance of aberrant QRS complexes when stimulated. In patients with paroxysmal SVT signs of sinus node dysfunction was detected in 6 patients, in the form of episodes of sinus arrhythmia (4 patients), migration pacemaker the atria (4 patients), sinoatrial blockade of II degree (3 patients), blockade of legs of bunch of gisa (2 patients), atrioventricular block degree II-III (1 patient), RTSNF more than 1500 MS in 1 patient, CRTSNF greater than 500 msec in 3 patients. Conclusion: in 6 of 9 patients with supraventricular paroxysmal tachycardia revealed signs of sinus node dysfunction, probably has a vagotonic in nature

Detection of Functional Significance of Coronary Stenoses Using Dynamic 13N-Ammonia Stress-PET/CT with Absolute Values of Myocardial Blood Flow and Coronary Flow Reserve

Objectives. The aim of the study was to compare the values of myocardial blood flow (MBF) at stress, MBF at rest and coronary flow reserve (CFR) obtained by 13Nammonia stress-PET/CT in patients with various degrees of coronary stenosis and in healthy patients. And thus to estimate the possible contribution of the stress-PET/CT quantitative data to the detection of functionally significant coronary stenoses in patients with coronary artery disease (CAD). Materials and methods. 63 patients (mean age 64±9 years) with known CAD underwent dynamic 13N-ammonia stress-PET/CT followed by calculation of MBF both at stress and at rest in absolute units and CFR. We compared quantitative values in two groups of patients with coronary artery stenosis: 1) ≥75% (n = 36) and 2) <75% (n = 27) confirmed by invasive coronary angiography and in group of healthy patients (n = 11). Results. MBF at stress was significantly lower in group with ≥75% diameter stenoses (median 1,44 [1,21; 1,85] mL/min per g) compared with group with <75% diameter stenoses (2,42 [1,75; 2,89] mL/min/g) and the normal group (2,54 [2,31; 2,86] mL/min/g), (p <0,001). There was no reliable difference in MBF at rest between the three groups (p = NS). CFR was significantly lower in the group of patients with severe ≥75% stenoses (1,85 [1,54; 2,31]) in comparison with patients group with stenoses of intermediate <75% severity (2,73 [2,19; 3,21]), and also in comparison with the normal group (3,12 [2,75; 3,23]), (p <0,001). Conclusion. The values of MBF at stress and CFR are significantly lower in patients with severe coronary arteries stenoses comparing with the group of patients with mild and moderate stenoses. The value of MBF at rest used independently has no diagnostic utility for detection of functional significance of coronary artery stenoses. Keywords: myocardial blood flow, coronary flow reserve, PET/CT, 13N-ammonia, coronary stenosis

Optical vortices and Airy spiral in chiral crystals

We consider both theoretically and experimentally propagation of a singular beam along a single chiral crystal and a stack of two such crystals with the oppo-site signs of their chirality coefficients. We develop a matrix approach for de-scribing behavior of the beam singularities. The beams with the eigen polariza-tion turn out to carry centered optical vortices with double topological charges. At the same time, a circularly polarized beam, when propagating, acquires an addi-tional unrequited phase having much to do with the geometrical Pancharatnam phase. The sign of the phase is defined by a direction of polarization circularity. To display experimentally the distribution of the geometrical phase we suggest employing two circularly polarized beams with opposite the circularities. The spiral image appearing behind the binary crystal system and the polarization fil-ter represents a set of spiral and ring edge dislocations of the wave front. They outline a profile of geometrical phase of the beam. Such the dislocation system is known in optical crystallography. It is nothing else than the four-fold Airy' spi-ral. We show also that the system of a chiral crystal with a purely anisotropic crystal, as well as a single chiral crystal combined with a circular polarization filter, are able to form a dark spiral line, too. However, such the line does not represent the edge dislocation and is shaped by means of a chain of optical vor-tices