The connection of the Degasperis-Procesi equation with the Vakhnenko equation

Travelling-wave solutions of the Degasperis-Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodicwave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well. Hone and Wang showed a connection between the DPE and the Vakhnenko equation (VE). Comparing the solutions of the DPE and the VE, we observe that, for both equations at interaction of waves, there are three kinds of phaseshift that depend on the ratio of wave amplitudes. In particular, there is a case when two interacted waves have phaseshifts in the positive direction

High-frequency soliton-like waves in a relaxing medium,

A nonlinear evolution equation is suggested to describe the propagation of waves in a relaxing medium. It is shown that for low-frequency approach this equation is reduced to the KdVB equation. The high-frequency perturbations are described by a new nonlinear equation. This equation has ambiguous looplike solutions. It is established that a dissipative term, with a dissipation parameter less than some limit value, does not destroy these looplike solutions

Loop-like Solitons

The physical phenomena that take place in nature generally have complicated nonlinear features. A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. One remarkable feature of the VE is that it possesses loop-like soliton solutions. Loop-like solitons are a class of interesting wave phenomena, which have been involved in some nonlinear systems. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE). The VPE can be written in Hirota bilinear form. The Hirota method not only gives the N-soliton solution but enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering transform (IST) method. This method is the most appropriate way of tackling the initial value problem (Cauchy problem). The standard procedure for IST method is expanded for the case of multiple poles, specifically, for the double poles with a single pole. In recent papers some physical phenomena in optics and magnetism are satisfactorily described by means of the VE. The question of physical interpretation of multivalued (loop-like) solutions is still an open question

Multi loop soliton solutions and their interactions in the Degasperis-Procesi equation

In this article, we construct loop soliton solutions and mixed soliton - loop soliton solution for the Degasperis-Procesi equation. To explore these solutions we adopt the procedure given by Matsuno. By appropriately modifying the $\tau$-function given in the above paper we derive these solutions. We present the explicit form of one and two loop soliton solutions and mixed soliton - loop soliton solutions and investigate the interaction between (i) two loop soliton solutions in different parametric regimes and (ii) a loop soliton with a conventional soliton in detail.Comment: Published in Physica Scripta (2012

Strain-induced kinetics of intergrain defects as the mechanism of slow dynamics in the nonlinear resonant response of humid sandstone bars

A closed-form description is proposed to explain nonlinear and slow dynamics effects exhibited by sandstone bars in longitudinal resonance experiments. Along with the fast subsystem of longitudinal nonlinear displacements we examine the strain-dependent slow subsystem of broken intergrain and interlamina cohesive bonds. We show that even the simplest but phenomenologically correct modelling of their mutual feedback elucidates the main experimental findings typical for forced longitudinal oscillations of sandstone bars, namely, (i) hysteretic behavior of a resonance curve on both its up- and down-slopes, (ii) linear softening of resonant frequency with increase of driving level, and (iii) gradual recovery (increase) of resonant frequency at low dynamical strains after the sample was conditioned by high strains. In order to reproduce the highly nonlinear elastic features of sandstone grained structure a realistic non-perturbative form of strain potential energy was adopted. In our theory slow dynamics associated with the experimentally observed memory of peak strain history is attributed to strain-induced kinetic changes in concentration of ruptured inter-grain and inter-lamina cohesive bonds causing a net hysteretic effect on the elastic Young's modulus. Finally, we explain how enhancement of hysteretic phenomena originates from an increase in equilibrium concentration of ruptured cohesive bonds that are due to water saturation.Comment: 5 pages, 3 figure

The inverse spectral problem for the discrete cubic string

Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-\phi'''=zm\phi$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for the case of $m$ being a finite positive discrete measure. In particular, explicit determinantal formulas for the measure $m$ are given. These formulas generalize Stieltjes' formulas used by Krein in his study of the corresponding second order ODE $-\phi''=zm\phi$.Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse Problems (http://www.iop.org/EJ/journal/IP

Organization of knowledge control on the discipline "Clinical immunology" for foreign students

Перевірка знань – найважливіший етап процесу навчання, у ході якого з'ясовуються повнота і якість знань іноземних студентів, прогалини і помилки в їхніх знаннях. Система організації контролю навчання іноземних студентів – це важливий крок у напрямі інтенсифікації й оптимізації навчально-виховного процесу у вищій школі.; Проверка знаний является важнейшим этапом процесса обучения, в ходе которого выясняются полнота и качество знаний иностранных студентов, пробелы и ошибки в их знаниях. Система организации контроля обучения иностранных студентов - это важный шаг в направлении интенсификации и оптимизации учебно-воспитательного процесса в высшей школе.; Сheck knowledge is the most important stage in the whole learning process, during which the completeness and quality of foreign students' knowledge, gaps and errors in their knowledge is clarified. The system of monitoring the education of foreign students is an important step in the direction of intensifying and optimizing the teaching and upbringing process in higher education

Localization of nonlinear excitations in curved waveguides

Motivated by the example of a curved waveguide embedded in a photonic crystal, we examine the effects of geometry in a ``quantum channel'' of parabolic form. We study the linear case and derive exact as well as approximate expressions for the eigenvalues and eigenfunctions of the linear problem. We then proceed to the nonlinear setting and its stationary states in a number of limiting cases that allow for analytical treatment. The results of our analysis are used as initial conditions in direct numerical simulations of the nonlinear problem and localized excitations are found to persist, as well as to have interesting relaxational dynamics. Analogies of the present problem in contexts related to atomic physics and particularly to Bose-Einstein condensation are discussed.Comment: 14 pages, 4 figure