Wave jumps in media described by a modified Schrödinger equation

The nonlinear Schrödinger equationA _t ±iA _xx+i‖A‖² A=0 describes an envelope of periodic waves with slowly varying parameters on water, in plasmas, and in nonlinear optics [1]. This equation can also be applied to steady periodic waves (the wave amplitude and wave number do not depend on time, the variablest andx are replaced by the variables of a horizontal coordinate system on the surface of the fluid [2]). In the present paper the properties of a modified Schrödinger equation involving the third and higher derivatives are studied. Solutions describing transition regions between uniform wave states are obtained numerically. If the structure of the transition region whose extent increases with time is not considered, these solutions may be interpreted as jumps.     

Model of ideal relaxation of thermoelastic stresses in the course of growth of single crystals

A simple dislocation model is proposed for relaxation of thermoelastic stresses generated during the growth of single crystals from a melt. This model does not require a solution of the kinetic equations for dislocations involved in relaxation and makes it possible to obtain the lower estimate of the dislocation density in the bulk of a grown crystal.     

Propagation of solitary waves over a semi-infinte underwater trough

Self-similar solutions describing the incidence of a uniform solitary wave on a semi-infinite linear trough are obtained on the basis of the nonlinear ray method [1]. Previously, in investigating the incidence of a wave on a trough [2] the conditions at the discontinuities present in the solutions were derived on the assumption that they are of low intensity. In the present study the use of the conditions at the discontinuities obtained by investigating soliton interaction [3–5] has made it possible to construct a series of new solutions and take into account wave reflection effects and the formation of a shadow zone beyond the trough. The types of solutions that occur are established in terms of the relations between the wave parameters and the relative depth of the trough. To ensure that self-similar solutions exist for all values of the parameters it was necessary to introduce a type of discontinuity not previously encountered [5–7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1987.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for discussing the work.     

Propagation of solitary waves over underwater ridges

The propagation of solitary waves is investigated on the basis of a nonlinear system of equations of hyperbolic type describing the motion of the crest of a solitary wave over the surface of a liquid of variable depth [1]. The existence of solutions with discontinuities, the boundary conditions at which are introduced on the basis of [2, 3], is assumed. In the case of an infinite cylindrical ridge both solitary and periodic captured waves are found. Depending upon the height of the ridge and the parameters of the wave, the encounter between a uniform wave and a semi-infinite ridge yields qualitatively different solutions — continuous and discontinuous, where the primary wave is broken down by the ridge into several solitary waves. The amplitude of the wave may either increase or decrease over the ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–93, January–February, 1985.The author is grateful to A. G. Kulikovskii and A. A. Barmin for their interest in his work, useful discussions and valuable comments offered during the preparation of the article for the press.     

Self-similar solutions describing the propagation of solitary waves above underwater ridges and troughs

The nonlinear ray method [1] is used to investigate the propagation of solitary waves over an uneven bottom. In the process of nonlinear evolution of the wave front, singular points develop in it; these are treated in the given model as discontinuities [2, 3]. In contrast to earlier studies, it is not assumed here that the intensity of the discontinuity is weak. Boundary conditions at the discontinuities are introduced on the basis of the results of Miles and Bakholdin [4–6], and this makes it possible to take into account the energy loss at a discontinuity and the effects of wave reflection and construct a number of new self-similar solutions for the propagation of a wave above a ridge and trough. The main attention is devoted to considering how the type of solution depends on the parameters of the wave and the relief. For certain values of the parameters, the self-similar solution of the encounter of a homogeneous wave with a ridge is not unique. The reason for this is the singularity of the relief at the end of the ridge. A numerical investigation has therefore also been made of the encounter of a wave with a ridge having a smooth relief at its end. For an under-water trough and a ridge—trough system, self-similar solutions with complete or partial reflection or transmission of the wave energy into the trough are found. A reflected wave can also arise from an encounter with a ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 137–144, September–October, 1985.I thank A. G. Kulikovskii and A. A. Barmin for their interest in the work and for valuable comments made as the paper was being prepared for press.     

Nonlinear resonances and wave jumps in media with higher-order dispersion

A new technique for systematically investigating biperiodic (two-wave) steady-state solutions is described with reference to modified Korteweg-de Vries and Schrödinger equations which generalize the conventional model equations for waves on water, in plasmas, and in nonlinear optics [1]. Among these solutions those with ordinary and resonance wave interactions are distinguished. Both singular solutions similar to the solitons of a resonantly interacting wave envelope and solitary waves are found. The soliton-like solutions obtained are used for describing the wave jump structure.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 113–124, July–August, 1996.     

Specific features of the morphology and distribution of gas inclusions in single-crystal sapphire ribbons grown by the Stepanov method

The distribution of gas inclusions in single-crystal sapphire ribbons with different crystallographic orientations, grown by the Stepanov method, is analyzed. In all ribbons gas inclusions are observed only in thin surface layers located at a depth of about 100 μm. One characteristic feature is the presence of several zones with different distributions and morphologies of inclusions. We observed inclusions of the two types: cylindrical pores and microbubbles. The differences in the morphology and zonality of the gas-inclusion distribution are consistent with the α-Al₂O₃ crystallography.     

Thermal field simulation and heat zone optimization for the growth of 50-mm basal-plane-faceted sapphire ribbons

The global heat transfer during growth of 50-mm basal-plane-faced sapphire ribbons in a cylindrical heat zone has been numerically simulated for different heat shield configurations. The ribbon thermoelastic strains were computated to estimate the heat zone quality. It is shown that shield adjustment and redistribution of radiative heat fluxes inside the heat zone make it possible to essentially decrease the thermal field curvature in the ribbon around the crystallization front and, therefore, decrease the thermoelastic strains by a factor of 2 to 3 (to 20 MPa).     

A Method of Video Goniography to Study the Morphology of the Lateral Surface of Profiled Monocrystals and the Device for Its Implementation

The method of video goniography and the device for its implementation has been described. The results of the study of the morphology of the lateral surface of profiled monocrystalline sapphire rods, which were grown by Stepanov’s method, of three main crystallographic orientations c{0001}, a{112̅0}, and m{101̅0} have been presented.

     

Analysis of Bulk Radiative Heat Transfer in a Crystal and in a Melt Using Numerical Simulation of the Sapphire Crystal Growth by the Stepanov Method

Technical Physics - The effect of bulk radiative heat transfer during the growth of profiled sapphire crystals from the melt is studied by the numerical simulation method. The peculiarities in the...