Probability Distributions of the Riemann Wave and an Integral of It

Doklady Physics - In this paper, we study the statistical characteristics of the Riemann wave and the integral of it. Such problems arise in the physics of fronts’ propagation, in particular,...     

Spatial Vector Solitons in Nonlinear Photonic Crystal Fibers

We study spatial vector solitons in a photonic crystal fiber (PCF) made of a material with the focusing Kerr nonlinearity. We show that such two-component localized nonlinear waves consist of two mutually trapped components confined by the PCF linear and the self-induced nonlinear refractive indices, and they bifurcate from the corresponding scalar solitons. We demonstrate that, in a sharp contrast with an entirely homogeneous nonlinear Kerr medium where both scalar and vector spatial solitons are unstable and may collapse, the periodic structure of PCF can stabilize the otherwise unstable two-dimensional spatial optical solitons. We apply the matrix criterion for stability of these two-parameter solitons, and verify it by direct numerical simulations.     

Spectral stability of periodic waves in the generalized reduced Ostrovsky equation

We consider stability of periodic travelling waves in the generalized reduced Ostrovsky equation with respect to co-periodic perturbations. Compared to the recent literature, we give a simple argument that proves spectral stability of all smooth periodic travelling waves independent of the nonlinearity power. The argument is based on the energy convexity and does not use coordinate transformations of the reduced Ostrovsky equations to the semi-linear equations of the Klein–Gordon type.     

Two-soliton interaction as an elementary act of soliton turbulence in integrable systems

Two-soliton interactions play a definitive role in the formation of the structure of soliton turbulence in integrable systems. To quantify the contribution of these interactions to the dynamical and statistical characteristics of the nonlinear wave field of soliton turbulence we study properties of the spatial moments of the two-soliton solution of the Korteweg–de Vries (KdV) equation. While the first two moments are integrals of the KdV evolution, the 3rd and 4th moments undergo significant variations in the dominant interaction region, which could have strong effect on the values of the skewness and kurtosis in soliton turbulence.     

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.     

Rogue internal waves in the ocean: Long wave model

Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.     

Thermally induced birefringence in Nd:YAG ceramics

Analytical expressions for eigenpolarizations and phase delays in grains of thermally loaded Nd:YAG ceramic rods have been derived. It is shown that the depolarization of radiation in polycrystalline ceramics results in beam modulation with a characteristic size of the order of the ceramic grain size. It is reasonable to increase the ratio of rod length to grain length both to diminish this modulation depth and to compensate for birefringence by use of known techniques.     

Averaging of nonlinearity-managed pulses

We consider the nonlinear Schrodinger equation with the nonlinearity management which describes Bose-Einstein condensates under Feshbach resonance. By using an averaging theory, we derive the Hamiltonian averaged equation and compare it with other averaging methods developed for this problem. The averaged equation is used for analytical approximations of nonlinearity-managed solitons.     

Jordan Superalgebras Defined by Brackets

Jordan superalgebras defined by brackets on associative commutativesuperalgebras are studied. It is proved that any such superalgebrais imbedded into a superalgebra defined by Poisson brackets.In particular, all Jordan superalgebras of brackets are i-special.The speciality of these superalgebras is also examined, andit is proved, in particular, that the Cheng–Kac superalgebrais special.