# Nonlinear Waves. Solitons

## The ring system with nonlinear elements, described by the two waves interaction model, manifesting the phenomena of complex analytical dynamics

This paper proposes a method of constructing of the ring system, in which the phenomena of complex analytical dynamics such as the Mandelbrot and Julia sets, are implemented in some approximation. The system is non-autonomous, includes frequency ?lters and nonlinear elements, described by the model of the resonant interaction of waves in quadratic nonlinear dispersive medium.

## Passage of the magnetostatic wave propagating through the lattice on the basis of the magnon crystal

The paper presents the results related to the construction of a model based on the coupled waves to describe the characteristics of magnetostatic waves passing through the structure on the basis of one-dimensional magnon crystal at various geometric parameters of the structure and at ?nite values of the input power.

## Модовая локализация в цепочках ферми–пасты–улама с произвольным порядком нелинейности

qБризеры – это точные периодические решения нелинейных акустических цепочечных систем, экспоненциально локализованные в модовом пространстве. Их наличие обусловливает динамическую локализацию энергии в исходно возбужденных модах и, как следствие, отсутствие термализации и сохранение линейчатого спектра. В данной работе исследуется вопрос о влиянии порядка степенной нелинейности g на длину локализации в q-пространстве, порог делокализации и масштабирование этих свойств с размером системы.

## Nonlinear electromagnetic wave passing through the layer with quadratic and fractionallypolynomial permittivity dependences on amplitude

The integral equations for powerful flat electromagnetic wave diffraction on nonlinear dielectric layer with cubic nonlinearity and fractionallypolynomial permittivity dependence on wave amplitude have been considered and solved. There are results which have been obtained by several numerical methods: series approaching, minimal discrepancy, power series expansion, and Runge–Kutt methods. Also the some analytical results are presented. The possibilities of power limiting, superexponential damping and some other effects in semiconducting plasma have bean shown by numerical simulation.

## Kink dynamics in the discrete klein–gordon model with asymmetric potential in the presence of ac driving

A discrete KleinGordon model with asymmetric potential that supports kinks free of the PeierlsNabarro potential (PNp) is constructed. Ratchet of kink under harmonic AC driving force is investigated in this model numerically and contrasted with the kink ratchet in the conventional discrete model where kinks experience the PNp. We show that the PNpfree kinks exhibit ratchet dynamics very much different from that reported for the conventional lattice kinks which experience PNp.

## Solitons in twofluid magnetohydrodynamics with nonzero electron inertia

The interaction between solitary waves in two fluid magnetohydrodynamic model of plasma with electron inertia taken into account is investigated analytically and numerically. Waves with linear polarization of a magnetic field are considered. A principal difference of the present work is the using of the «exact» equations, instead of the modeling equations. It is numerically shown, that solitary waves really are solitons, i.e. their interaction is similar to interaction of colliding particles.

## Characteristics of gap discrete breathers in crystals with nacl structure

Molecular dynamics method is used to study the effect of mass ratio of anions and cations on the phonon spectra of the crystal with NaCl structure and on the discrete breathers existence conditions and properties of gap discrete breathers. We show that discrete breathers can be easily excited for the mass ratio less than 0.2, when the gap in the phonon spectrum is wide enough to support them. When the mass ratio is equal to 0.1 we could find at least three types of stable discrete breathers, differed by the number of large amplitude atoms and by polarization of oscillation.

## Introduction to discrete breathers theory

We make a basic review of the theory of discrete breathers – spatially localized solutions in nonlinear lattices. We describe the mathematical conditions and physical prerequisites of their existence and methods of their study by example of one-dimensional lattices. We consider localized solutions with infinite and finite lifetimes. We include some new results within the problems of discrete breather generation resulting from harmonic wave destruction and controlling the formation of rotational breather solutions by external forcing.

## The leaky modes of multilayered waveguide with nonlinear dielectrics

Characteristics of the leaky modes, propagating along the planar layered waveguides with nonlinear media, are studied. The mode phase constants and attenuation coe?cients are calculated. In nonlinear structures the dependencies of the mode ?eld distributions on the longitudinal coordinate are shown to di?er from exponential ones which are typical for the linear problems. This property results in the e?ect of the di?erent modes transformation even for the regular geometry of the waveguide.

## Magnetostatic surface waves parametric instability in two-dimensional (2d) magnonic crystals

First order (three-magnon) parametric instability of magnetostatic surface waves (MSSW) was experimentally studied in two-dimensional (2D) magnonic crystals with rhombic and square lattices with lattice parameter 37–40 mm. The instability was produced by etching of holes 32 mm in diameter and 1–2 mm in depth in the 16 mm-thick yttrium iron garnet (YIG) ?lm. It was found, that MSSW threshold powers for parametric instability development in case of 2D magnonic crystals are of the order of two times greater than analogous threshold values for starting YIG ?lms.