40 research outputs found
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 -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 on the real line or a finite interval, the "cubic string"
is the third order ODE where 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
being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE .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