Lyapunov stability of singular planar systems related to dispersion-managed solitons in optical fiber

In this paper, we consider two singular planar differential systems which can describe the evolution of the optical pulse width and chirp for the so-called dispersion-managed solitons. Based on the method of third order approximation in combination with some quantitative information obtained by the upper-lower solutions method and the averaging method, some results on the existence and Lyapunov stability of the periodic solutions are obtained. Moreover, the formula of the first twist coefficient and a stability criterion of a nonlinear differential equation are also established.

Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

This work is one of a series of papers that is devoted to the further investigation of polynomial splines and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. In some cases, the use of the trigonometric approximations is preferable to the polynomial approximations. Here we continue to compare these two types of approximation. The Lebesgue functions and constants are discussed for the polynomial splines and the trigonometric splines. The examples of the applications of the splines to image enlargement are given.

Head-on collision between capillary–gravity solitary waves

The present study deals with the head-on collision process between capillary–gravity solitary waves in a finite channel. The present mathematical modeling is based on Nwogu’s Boussinesq model. This model is suitable for both shallow and deep water waves. We have considered the surface tension effects. To examine the asymptotic behavior, we employed the Poincaré–Lighthill–Kuo method. The resulting series solutions are given up to third-order approximation. The physical features are discussed for wave speed, head-on collision profile, maximum run-up, distortion profile, the velocity at the bottom, and phase shift profile, etc. A comparison is also given as a particular case in our study. According to the results, it is noticed that the free parameter and the surface tension tend to decline the solitary-wave profile significantly. However, the maximum run-up amplitude was affected in great measure due to the surface tension and the free parameter.

Interval estimation of trigonometrical splines

Quite often, it is necessary to quickly determine variation range of the function. If the function values are known at some points, then it is easy to construct the local spline approximation of this function and use the interval analysis rules. As a result, we get the area within which the approximation of this function changes. It is necessary to take into account the approximation error when studying the obtained area of change of function approximation. Thus, we get the range of changing the function with the approximation error. This paper discusses the features of using polynomial and trigonometrical splines of the third order approximation to determine the upper and lower boundaries of the area (domain) in which the values of the approximation are contained. Theorems of approximation by these local trigonometric and polynomial splines are formulated. The values of the constants in the estimates of the errors of approximation by the trigonometrical and polynomial splines are given. It is shown that these constants cannot be reduced. An algorithm for constructing the variation domain of the approximation of the function is described. The results of the numerical experiments are given.

MODULATION STABILITY OF WAVE PACKETS IN A THREE-LAYER HYDRODYNAMIC SYSTEM

The article is devoted to the problem of propagation of weakly nonlinear wave-packets along contact surfaces in a three-layer hydrodynamic system "half space – layer – layer with rigid lid". The condition of solvability of the problem in the third-order approximation is obtained, the evolution equation is derived in the form of a nonlinear Schrodinger equation and the modulation stability condition for its solutions is obtained. The stability diagram and its analysis are presented for the solution which takes place in the case of the balance between dispersion and non-linearity.

Tunnelling Mechanism in Noncommutative Space with Generalized Uncertainty Principle and Bohr-Like Black Hole

The paper deals with nonthermal radiation spectrum by tunnelling mechanism with correction due to the generalized uncertainty principle (GUP) in the background of noncommutative geometry. Considering the reformulation of the tunnelling mechanism by Banerjee and Majhi, the Hawking radiation spectrum is evaluated through the density matrix for the outgoing modes. The GUP corrected effective temperature and the corresponding GUP corrected effective metric in noncommutative geometry are determined using Hawking’s periodicity arguments. Thus, we obtain further corrections to the nonstrictly thermal black hole (BH) radiation spectrum which give new final distributions. Then, we show that the GUP and the noncommutative geometry modify the Bohr-like BH recently discussed in a series of papers in the literature. In particular, we find the intriguing result that the famous law of Bekenstein on the area quantization is affected neither by noncommutative geometry nor by the GUP. This is a clear indication of the universality of Bekenstein’s result. In addition, we find that both the Bekenstein-Hawking entropy and the total BH entropy to third-order approximation are still functions of the BH quantum level.

Asymmetric shape transitions of epitaxial quantum dots

We construct a two-dimensional continuum model to describe the energetics of shape transitions in fully faceted epitaxial quantum dots (strained islands) via minimization of elastic energy and surface energy at fixed volume. The elastic energy of the island is based on a third-order approximation, enabling us to consider shape transitions between pyramids, domes, multifaceted domes and asymmetric intermediate states. The energetics of the shape transitions are determined by numerically calculating the facet lengths that minimize the energy of a given island type of prescribed island volume. By comparing the energy of different island types with the same volume and analysing the energy surface as a function of the island shape parameters, we determine the bifurcation diagram of equilibrium solutions and their stability, as well as the lowest barrier transition pathway for the island shape as a function of increasing volume. The main result is that the shape transition from pyramid to dome to multifaceted dome occurs through sequential nucleation of facets and involves asymmetric metastable transition shapes. We also explicitly determine the effect of corner energy (facet edge energy) on shape transitions and interpret the results in terms of the relative stability of asymmetric island shapes as observed in experiment.

Analysis of Acoustic-Structure Interaction Using a High-Order Doubly Asymptotic Approximation

In this paper, a new third-order approximation model for an acoustic-structure interaction problem is introduced. The new approximation model is designed to be an accurate and a stable model for predicting the response of a submerged structure. The proposed model is obtained by combining two lower order approximation models instead of using an operator matching method. The stability of this model is checked by a modal analysis. Finally, the approximation model is coupled to the spherical shell structure, and its performance is checked by a shock analysis.