Traveling waves in a nonlinear medium with cubic nonlinearity

Unlike linear nondispersive media, which allow propagation of wave packets of arbitrary forms, nonlinear media admit only certain profiles of traveling waves. Here we examine media with Duffing oscillators, i.e., with bound electrons for which an equilibrium disturbance causes forces proportional to the first and third powers of deviation. We show that the linearly polarized traveling plane waves such media can transmit have profiles modulated as Jacobi elliptic functions. When discussing propagation across an interface between different media, only incidence from the side of the linear medium is considered. Even in this case, to launch a traveling wave in the nonlinear medium, a severe restriction must be imposed on the incident wave’s amplitude.     

Shear standing waves in a resonator with a rubberlike inhomogeneous medium

A modification of the finite-element method is proposed for calculating shear standing waves in a resonator filled with an incompressible elastic medium with allowance for the finite dimensions of the resonator and inhomogeneities of the shear modulus. Resonance curves are calculated for resonators with inhomogeneities in the form of cavities and elastic inclusions. Numerical calculations are compared with experimental data.     

Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a “breathing” wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of “unpredictable” behavior of the breather of the Gardner equation is discussed.     

Optical pulsing in a resonator with a highly dispersive nonlinearity

By introducing an intracavity Doppler shift in a resonator with a highly dispersive nonlinear medium, a train of optical pulses is generated whose features are related to the slow/fast-light response of the medium. The cavity transmission is asymmetric and the pulse shape is modified differently depending on the direction of the Doppler shift, hence, on the sign of the group delay provided by the dispersive process.     

Nonlinear equations for quasi-monochromatic waves near a caustic in a dispersive dissipative medium with cubic or quadratic nonlinearity

The behavior of a quasi-monochromatic nonlinear wave near a caustic is considered. Nonlinear ordinary differential equations for a dispersive dissipative medium with a cubic or quadratic nonlinearity are derived. For the latter medium, nonstationary equations describing it near the caustic are presented with allowance for the dissipative dispersive terms. These equations yield ordinary ones for quasi-monochromatic waves. The amplitude of the second harmonic is expressed in terms of the squared amplitude of the first harmonic. The amplitude of the second harmonic, as well as the solution as a whole, increases near the caustic.     

Self-gravitating scalar field with cubic nonlinearity

For a self-gravitating massless conformally invariant scalar field a solution is obtained to the Einstein equations for which the geometry of space-time remains arbitrary. For a scalar field with cubic nonlinearity, a static solution to the Einstein equations possessing plane symmetry is found. A cosmological model with nonlinear scalar field in the class of conformally flat Friedmann metrics is investigated. An example is given of an exact solution to the equations of the gravitational field with singularity in the infinite past.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 18–22, December, 1980.     

Self-modulation of Bessel-Gaussian wave beams in a medium with cubic nonlinearity

A semi-discrete dynamic model has been developed for the formation of the spatial structure of wave fields in a medium with cubic nonlinearity. The characteristic features of self-focusing and conical modulation of intense Bessel-Gaussian light beams of different orders have been studied in different stages of their evolution during propagation. It has been shown that as a result of nonlinear refraction, in the far zone wave structures are formed consisting of three spatially separated conical beams. Increasing the cone angle of the wave vectors leads to a decrease in the effect of conical modulation of the radiation, and improves the structural stability of the beam. The considered self-modulation effects can be used for passive limiting of the laser radiation power. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 5, pp. 626–630, September–October, 2006.     

Surface waves in a crystal with switching Kerr nonlinearity

In this paper, we explore transverse electric surface waves propagating along the crystal with jump change of Kerr nonlinearity in dependence on field amplitude. The dielectric permittivity in the proposed model of nonlinearity is characterized by abruptly changing unperturbed dielectric constant and Kerr nonlinearity coefficient from one value to another when field amplitude exceeds the threshold value of the switching field. This allows to find exact solutions of model equations in different cases of nonlinearity signs, and to obtain the dependence of wave characteristics, including total power flux, on effective refractive index in explicit form. Such solutions describe two new types of nonlinear surface waves with specific structure depending on electric field amplitude. We derive the conditions of surface domain formation. It is found that the largest percentage of radiation is concentrated within the domain.     

Nonlinear standing waves in a resonator with feedback control

An experimental study is presented to demonstrate that nonlinear effect on standing waves in a resonator can be reduced by a feedback loop responding to the second harmonic. The resonator was a cylindrical tube sealed at one end and driven by a horn driver unit at another end. The feedback control loop consisted of a pressure sensor, a frequency filter, a phase shifter, and an actuator. The results show that the waveform distortions can be eliminated and large amplitude sinusoidal pressure oscillations are obtained. A simple model is proposed for a qualitative discussion on the control mechanism, which shows that the feedback loop alters the imaginary part of the complex mode frequency so as to suppress (or enhance) the second harmonic.     

Modes in the Fabry-Perot resonator with the Kerr nonlinearity

Summary The symmetric modes in the cavity of Fabry-Perot type filled by the medium with the Kerr nonlinearity are constructed. The field appears to be double periodic along the longitudinal axis.

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Riassunto Si costruiscono i modi simmetrici nella cavità del tipo di Fabry-Perot riempita dal mezzo con la non linearità di Kerr. Il campo risulta doppiamene periodico lungo l’asse longitudinale.

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Резюме Конструируются симметричные моды в резонаторе Фабри-Перо, заполненном средой с нелинейностью Керра. Оказывается, поле имеет двойную периодичность вдоль продольной оси.

     

Diffraction of X-ray spatially inhomogeneous beam in a crystal with cubic nonlinearity

Diffraction of a spatially inhomogeneous X-ray wave was theoretically studied in a crystal with cubic nonlinear response to the strength of an external electric field. Using numerical calculations in case of two-beam diffraction of a narrow incident beam, the intensity distributions on the output surface of the crystal was investigated depending on the thickness and intensity of the incident beam. The results of numerical calculations of integral (spatial) coefficients of transmission and reflection are given as functions of incident wave intensity for fixed thickness of the crystal.     

Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion

Abstract

The vector-coupled nonlinear Schrödinger equation, which can be applied to describe the propagation of Thirring optical solitons in birefringent ?bers with Kerr law nonlinearity, detuning, intermodal dispersion and spatiotemporal dispersion, has been studied analytically. By means of the complex envelope function ansatz, exact Thirring bright-dark combosolitons are reported, and the properties of these solitons are discussed.     

Dispersive shock waves with nonlocal nonlinearity

We consider dispersive optical shock waves in nonlocal nonlinear media. Experiments are performed using spatial beams in a thermal liquid cell, and results agree with a hydrodynamic theory of propagation.     

Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity

Propagation of Gaussian beams in the negative-index metamaterials (NIMs) with cubic nonlinearities is investigated, both theoretically and numerically. The role of the status of the incident Gaussian beam, which is scaled by a converging parameter in this paper, in beam self-focusing and self-defocusing in NIMs is specially identified. The expressions for beam self-focusing and self-defocusing for different converging parameter cases, and the dependence of the critical power and the focus location of self-focusing in NIMs on the converging parameter are obtained. It is found that it is the divergent rather than convergent incident beams which are self-focused more quickly in NIMs with defocusing nonlinearities, in sharp contrast with the propagation property of Gaussian beams in conventional Kerr media, in which beam self-focusing only occurs in the media with focusing nonlinearities and a convergent incident beam self-focuses more quickly than a divergent one. By adjusting the converging parameter of incident Gaussian beam or the controllable magnetic permeability of NIM, or both, one can manipulate the beam self-focusing in NIMs at will.     

On cubic nonlinearity of cracked media

An equation of state with cubic nonlinearity has been obtained for the model of a cracked medium. It is shown that cracks in a solid can lead not only to a strong change of the cubic nonlinearity parameter but also to the sign reversal of the latter. We describe the experiment on the measurement of a nonlinear shift in the resonance frequency of a bar resonator of glass with cracks where the positive shift of the resonance frequency was observed. We estimate the effective parameter of the cubic nonlinearity for such a resonator by comparing theoretical and experimental dependences. Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 7, pp. 897–902, July. 1997.     

A three-component Camassa-Holm system with cubic nonlinearity and peakons

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions. Additionally, reductions of the 3CH system are investigated so that a new integrable perturbed CH equation with cubic nonlinearity is generated to possess peakon solutions.     

Entangled states of light during its propagation in a transparent medium with cubic nonlinearity

When two coherent waves propagate in a transparent medium with cubic nonlinearity, a nonclassical state appears which is characterized by the quantum correlation of the waves. On the basis of exact solutions to the problem, it was found that such states lead to violation of the separability criterion for continuous variables.