Discontinuities of the variables that characterize the propagation of solitary waves in a fluid layer

A nonlinear system of equations of hyperbolic type describing the propagation of solitary waves is considered [1]. A solitary wave is characterized in this approximation by two variables — the energy density per unit length measured along its crest, and the direction of the normal to the wave crest. The evolution of a wave described by the system may lead to the appearance of discontinuities, at which there are jumps in the energy density and the direction of the wave crest [2]. To establish the conditions at the discontinuities, a solution describing the interaction of nonparallel solitons [3, 4] is used. The obtained conditions are used to solve the problem of the decay of an arbitrary discontinuity in terms of soliton variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–93, May–June, 1984.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and valuable comments in the preparation of the paper.     

Fine structure of faces and imperfection of surface layers of shaped sapphire crystals

The development of faces on cylindrical and ribbon-shaped sapphire single crystals grown from a melt using the Stepanov method is investigated by optical and electronic microscopy. The crystallographic indices on the detected faces are established, and the microstructure of the growth layers is studied. The X-ray spectral microanalysis of foreign particles on the sapphire surface confirms the presence of molybdenum single crystals on it. Cathodoluminescence spectra of surface layers of sapphire single crystals are obtained.     

Study of the real structure of basal-plane-faceted sapphire ribbons

The structure of basal-plane-faceted sapphire ribbons with cross sections of 30 × 1.5 and 50 × 1.5 mm, grown by the Stepanov method, has been investigated using X-ray diffraction topography and selective etching. The model of block formation in such ribbons through interaction of dislocations of prismatic slip system is confirmed. The role of gas bubbles as stress concentrators and dislocation sources is established. It is shown that the dislocations of basal-slip system are involved in the development the low-angle boundaries and block formation.     

Redistribution of thermoelastic stresses in shaped crystals: Numerical simulation

The shape of crystals was considered for the first time as an important factor controlling thermoelastic stresses during crystal growth. It was found on the basis of numerical simulation that thermoelastic stresses can be redistributed by varying the shape of a crystal during growth. In this way, large weakly stressed regions can be obtained in the crystal owing to the formation of locally stressed regions. As an example, a model of a sapphire ribbon (50 × 175 mm²) divided by waists into three plates (50 × 50 mm²) is discussed. Different crystallographic orientations and slip systems are considered.     

Modeling habit forms of sapphire crystals using the principles of periodic-bond-chain method

The potential of the periodic-bond-chain method for calculating the sequence of manifestation of faces of corundum single crystals is considered. The leading role of the faces of the pinacoid, high rhombohedron, and hexagonal prism is demonstrated. The calculation results are compared with the experimental data on faceting the lateral surface of cylindrical sapphire single crystals grown by the Stepanov method and with the faceting data for crystals grown by the flux method and natural crystals.

     

Calculation of thermoelastic stresses near the crystallization front for melt-grown single-crystal rods with a circular cross section

Approximate expressions for the thermoelectric stress tensor components in an isotropic crystal rod of a circular cross section, applicable in the entire crystal including the region near the crystallization front, are obtained. Using the resultant approximate formulas, the stress fields in leucosapphire single crystals are calculated for model temperature fields. It is shown that exactly near the crystallization front, thermoelastic stresses reach maximal values.     

The influence of thermal screens on the temperature distribution,thermal stress,and defect structure during growth of shaped sapphire crystals

The ways in which a block structure is formed in shaped sapphire single crystals grown from melt by the Stepanov method are considered. The measured temperature distributions and results of a mathematical modeling of the heat exchange in the growth zones, as well as the calculated thermoelastic fields and measured residual stresses, are reported. The possibility of effectively controlling the thermal fields and growth of block-free crystals by choosing optimal screening is shown for single crystals in the form of tubes and basal-plane-faceted ribbons.     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

Study of magnetosonic solitary waves for the electron magnetohydrodynamics equations

Electron magnetohydrodynamics equations are derived with allowance for nonlinearity, dispersion, and dissipation caused by friction between the ions and electrons. These equations are transformed into a form convenient for the construction of a numerical scheme. The interaction of codirectional and oppositely directed magnetosonic solitary waves with no dissipation is computed. In the first case, the solitary waves are found to behave as solitons (i.e., their amplitudes after the interaction remain the same), while, in the second case, waves are emitted that lead to decreased amplitudes. The decay of a solitary wave due to dissipation is computed. In the case of weak dissipation, the solution is similar to that of the Riemann problem with a structure combining a discontinuity and a solitary wave. The decay of a solitary wave due to dispersion is also computed, in which case the solution can also be interpreted as one with a discontinuity. The decay of a solitary wave caused by the combined effect of dissipation and dispersion is analyzed.     

Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation

A numerical solution of a cubic Schrödinger equation established the phenomenon of Mach reflection of Stokes waves from a vertical wall [1]. In [2], this phenomenon was interpreted as the resonance interaction of three waves. This was based on the derivation of averaged equations of the interaction of two waves, the region of interaction of the incident and reflected waves being treated as a self-similar solution of these equations. The present paper establishes the possibility of describing these solutions by the relations of three-wave resonance; the mathematical significance of the resonance as splitting into two waves is revealed; and the properties of the averaged system are investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–116, January–February, 1992.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and also colleagues at the Mechanics Section of the V. A. Steklov Mathematics Institute of the USSR Akademy of Sciences for critical comments that stimulated interest in the paper.