Structure and stability of quasiparallel small-amplitude magnetohydrodynamic shocks

The structure and stability of quasiparallel magnetohydrodynamic shock waves of small but finite amplitude are investigated. Only those waves whose propagation velocities are close to the Alfvén velocity are considered, i.e., fast shock waves in a medium in which the Alfvén velocity is greater than the speed of sound and slow shock waves in a medium in which the Alfvén velocity is less than the speed of sound and, moreover, intermediate (nonevolutionary) shock waves.In conclusion, the author wishes to thank A. A. Barmin for discussing his results and offering useful comments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 153–160, July–August, 1989.     

Oblique interaction of alfvén and contact discontinuities in magnetohydrodynamics

A general method of solving problems of the interaction of stationary discontinuities is proposed. The problem of the oblique incidence of an Alfvén plane-polarized discontinuity on a contact discontinuity is examined in the general formulation. A solution is constructed numerically over the entire range of variation of the governing parameters. A number of effects associated with the magnetohydrodynamic nature of the interaction are explored. For example, the formation in space of sectors in which the density falls by several orders (almost to a vacuum) is detected. The solutions obtained are of interest, for example, for investigating the interaction between Alfvén discontinuities in the solar wind and the magnetopause, plasmopause and other inhomogeneities whose boundary can be approximated by a contact discontinuity [13–15].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 131–142, January–February, 1990.     

Nonlinear internal waves in a fluid of infinite depth in the presence of a horizontal magnetic field

An investigation is made into the propagation of long nonlinear weakly nonone-dimensional internal waves in an incompressible stratified fluid of infinite depth in the presence of a horizontal magnetic field. It is shown that such waves are described by an equation representing the extension of the Benjamin-Ono equation to the weakly nonone-dimensional case. The equation obtained differs from that obtained in [4], which is attributable to the anisotropy of the medium resulting from the presence of a magnetic field. The stability of a soliton with respect to flexural perturbations is investigated. A particular case of the variation of the density with height at constant Alfvén velocity is examined in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 65–72, November–December, 1987.     

Motion of an incompressible conducting liquid in a strong self-consistent magnetic field

One of the common regimes of operation of many laboratory and industrial magnetohydrodynamic (MHD) devices using liquid metals as working medium is the regime for which the Alfvén number A, the ratio of the magnetic and kinetic energy densities, appreciably exceeds unity. For example, for a typical MHD device [1] with characteristic length 0.1 m of the working region, velocity 1 m/sec of the medium, and magnetic induction 1 T (the medium is molten sodium at temperature 330°C) the Alfvén number is A - 900. To simplify the investigation of the processes in such devices, one can use the approximation of a strong magnetic field proposed by Somov and Syrovatskii [2] to describe certain types of hydrodynamic flows of a dissipationless plasma in a magnetic field. In the present paper, the approach to the analysis of the self-consistent magnetohydrodynamic problem in this asymptotic approximation is extended to the case of an incompressible liquid with finite conductivity. A study is made of the closed reduced system of MHD equations obtained from the complete model in the zeroth order in the small parameter A^–1, in which the magnetic field is a force-free field. An investigation is made of the free diffusion of force-free magnetic field with constant coefficient a of proportionality between the current density and the magnetic induction in a spatially unbounded liquid, and the kinematic properties of a velocity field of the liquid in which the force-free nature of the magnetic field is maintained during the damping process are determined. It is shown that the complete class of such velocity fields is represented by the group of rigid-body motions of the liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–9, January–February, 1991.     

Three-dimensional steady flows of stratified fluid and internal waves

A study is made of three-dimensional steady flows of an ideal heavy incompressible fluid stratified in each layer over a flat or asymptotically flat base. Mixed Euler-Lagrange variables are chosen in which surfaces of constant density, including the layer division boundaries, become flat and parallel to the plane of the base. The original problem is reduced to a nonlinear boundary-value problem for a system of three quasilinear equations in a plane layer. This system of equations is used to construct an asymptotic theory of long waves in the three-dimensional case, which has particular solutions in the first approximation in the form of solitons and soliton systems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 127–132, May–June, 1985.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

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.     

Damping of solitons in bubble media

The author considers the problem of damping of solitons in a bubble medium because of viscosity and interphase heat and mass transfer nonequilibrium on the assumption that their evolution is described by successive interchange of steady soliton solutions. The changes in the amplitude of the solitons obtained from analytical expressions are compared with the available experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–61, July–August, 1987.     

Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Stability of solitary waves in dispersive media described by a fifth-order evolution equation

The problem of the existence and dynamical stability of solitary wave solutions to a fifth-order evolution equation, generalizing the well-known Korteweg-de Vries equation, is treated. The theoretical framework of the paper is largely based on a recently developed version of positive operator theory in Fréchet spaces (which is used for the existence proof) and the theory of orbital stability for Hamiltonian systems with translationally invariant Hamiltonians. The validity of sufficient conditions for stability are established. The shape of solitary waves under analysis are determined by a numerical solution of the boundary-value problem followed by a correction using the Picard method of 4–12 orders of accuracy.     

On the theory of solitons in systems with dissipation

In the weakly nonlinear approximation wave processes in flowing films, the propagation of concentration waves in chemical reactions, the hydrodynamic instability of a laminar flame, and thermocapillary convection in a thin layer are described by equations of the type h_t + 4hh_x + h_xx + h_xxxx=0. A special role in wave processes is played by nonlinear localized signals-solitary waves or solitons. In this paper the methods of the theory of dynamical systems are used to carry out a full investigation of solutions of the stationary soliton type for the above-mentioned equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 91–97, May–June, 1986.     

Level kegulation of ground water in irrigation

Level regulation of ground water is important for preventing the irrigated ground from becoming bogged up or salinated; the evaporation and the existence of a weakly permeable horizontal waterproof stratum are taken into account. The solution is found in an explicit form. It is also shown that the solution tends asymptotically either to one of the two stationary solutions or to periodic solutions which are also obtained in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 125–133, September–October, 1973.     

Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System

We prove the existence of locally unique, symmetric standing pulse solutions to homogeneous and inhomogeneous versions of a certain reaction–diffusion system. This system models the evolution of photoexcited carrier density and temperature inside the cavity of a semiconductor Fabry–Pérot interferometer. Such pulses represent the fundamental nontrivial mode of pattern formation in this device. Our results follow from a geometric singular perturbation approach, based largely on Fenichel's theorems and the Exchange Lemma.     

Generation of compressible modes in MHD turbulence

Astrophysical turbulence is magnetohydrodynamic (MHD) in nature. We discuss fundamental properties of MHD turbulence and in particular the generation of compressible MHD waves by Alfvénic turbulence and show that this process is inefficient. This allows us to study the evolution of different types of MHD perturbations separately. We describe how to separate MHD fluctuations into three distinct families: Alfvén, slow, and fast modes. We find that the degree of suppression of slow and fast modes production by Alfvénic turbulence depends on the strength of the mean field. We review the scaling relations of the modes in strong MHD turbulence. We show that Alfvén modes in compressible regime exhibit scalings and anisotropy similar to those in incompressible regime. Slow modes passively mimic Alfvén modes. However, fast modes exhibit isotropy and a scaling similar to that of acoustic turbulence both in high and low plasmas. We show that our findings entail important consequences for star formation theories, cosmic ray propagation, dust dynamics, and gamma ray bursts. We anticipate many more applications of the new insight to MHD turbulence and expect more revisions of the existing paradigms of astrophysical processes as the field matures. PACS 47.65.+a; 52.30.Cv; 52.35.Ra; 95.30.Qd     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Field-aligned flow of a conducting fluid past a source of magnetism

Summary A prescribed source of magnetism moves at constant speed through a viscous conducting incompressible fluid with an aligned uniform magnetic field. The velocity and magnetic fields induced at a distance from the source are calculated. The induced fields are also calculated for the case in which the applied field is absent. Although no special symmetry or alignment is assumed, the source is ideal in the sense that enclosures (wires or magnets) are infinitesimal in at least two dimensions. Dynamical interactions will occur in a viscous fluid and their effect in the far field is estimated.As a consequence of finite conductivity and viscosity, the usual wakes are present which trail or lead the source depending upon the sign of (1–A ²), where A is the ratio of the source speed to the Alfvén speed in the undisturbed fluid. Outside the wake the total perturbation magnetic field due to the source is the static field plus a monopole field, divided by (1–A ²).An estimate is also made of the rate at which energy is dissipated as a consequence of viscous interactions and ohmic heating throughout the fluid, outside the immediate vicinity of the source.Geo-Astrophysics Laboratory.Plasma Physics Laboratory.     

Bright hump solitons for the higher-order nonlinear Schrödinger equation in optical fibers

Under investigation is the higher-order nonlinear Schrödinger equation with the third-order dispersion (TOD), self-steepening (SS) and self-frequency shift, which can be used to describe the propagation and interaction of ultrashort pulses in the subpicosecond or femtosecond regime. Through the introduction of an auxiliary function, bilinear form is derived. Bright one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. From the one-soliton solutions, we present the parametric regions for the existence of single- and double-hump solitons, and find that they are affected by the coefficients of the group velocity dispersion (GVD) and TOD. Besides, propagation of the one single- or double-hump soliton is observed. We analytically obtain the amplitudes for the single- and double-hump solitons, and calculate the interval between the two peaks for the double-hump soliton. Moreover, soliton amplitudes are related to the coefficients of the GVD, TOD and SS, while the interval between the two peaks for the double-hump soliton is dependent on the coefficients of the GVD and TOD. Interactions are seen between the (i) two single-hump solitons, (ii) two double-hump solitons, and (iii) single- and double-hump solitons. Those interactions are proved to be elastic via the asymptotic analysis.     

Some one-dimensional unsteady adiabatic gas flows with plane symmetry

A method for the approximate analytic investigation of one-dimensional adiabatic gas flows in the form of arbitrary small perturbations of a simple wave is proposed. A class of exact solutions which, in particular, describes the flows arising from the short intense impact of a piston moving under gas pressure is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 96–104, September–October, 1986.     

Nonstationary waves in a film of viscous liquid

A numerical investigation is made of the development of initial perturbations in a thin film of viscous liquid. It is shown that the resulting wave structure passes through complicated intermediate shapes such as solitons in which there are elevations and depressions of the surface, At large times, a wave regime is formed that is close to the optimal regime with respect to the wave number. The picture of the intermediate wave forms depends on the initial data, while the final result of the development depends weakly on the initial data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–154, May–June, 1981.We thanks A. A. Bunov for assistance in the calculations.     

Propagation of long nonlinear waves in a ponderable fluid beneath an ice sheet

In Sec. 1 the stability of small-amplitude steady-state periodic solutions of Eq. (0.1) in the neighborhood of k=k_n are investigated. The results of the investigations are consistent with those of [1]. In Sec. 2 the stability of periodic waves not lying in the neighborhood of resonance is considered. It is shown that in the region of instability when =1 steady-state solutions of the soliton type with oscillatory structure may exist. In Sec. 3 the properties of certain exact solutions — periodic waves and solitons — are studied in relation to the nature of the singular points of the dynamical system derived from (0.1). In Sec. 4 the evolution of rapidly decreasing Cauchy data is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1989.