Isometric imbedding in E3 of noncompact domains of nonpositive curvature

Author's abstract of a dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on Oct. 21, 1977, at a session of the Specialized Council D 002.38.02 at the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. Official opponents: Academician of the Academy of Sciences of the USSR, Doctor of Physicomathematical Sciences A. V. Pogorelov; Professor, Doctor of Physicomathematical Sciences É. G. Poznyak; Professor, Doctor of Physicomathematical Sciences A. S. Fedenko.Translated from Matematicheskie Zametki, Vol. 25, No. 5, pp. 785–797, May, 1979.     

Exact self-consistent plane-symmetric solutions of the spinor-field equation with a nonlinear term that depends on theS 2−P 2 invariant

Exact plane-symmetric solutions of the spinor-field equation with zero mass parameter and nonlinear term that depends arbitrarily on the S²−P² invariant are derived with consideration of an intrinsic gravitational field. The existence of regular solutions with localized energy density among the solutions obtained is investigated. Equations with powerlaw and polynomial nonlinearity types are examined in detail. For the power-law nonlinearity, when the nonlinear term entering into the Lagrangian has the form L_N=γIⁿ, where γ is the nonlinearity parameter and n=const, it is shown that the initial system of Einstein and spinor-field equations has regular solutions with localized energy density only under the conditions λ=−Λ² < 0, n > 1. In this case, the examined field configuration posesses a negative energy. In the case of polynomial nonlinearity, regular solutions with localized energy density T ₀ ⁰ (x), positive energy (upon integration over y and z between finite limits), and an everywhere regular metric that transforms into a two-dimensional space-time metric at spatial infinity are obtained. It is shown that the initial nonlinear spinor-field equations in two-dimensional space-time have no solutions with localized energy density. Thus, it is established that the intrinsic gravitational field plays a regularizing role in the frmation of regular localized solutions to the examined nonlinear spinor-field equations. Russian University of People's Friendship. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 12–19, November, 1999.     

On the regularity of oricyclic coordinates

Suppose there is defined in the plane a complete metric W^?, whose curvature K satisfies the inequality ?k ₂ ² ≤ K≤ ?k ₁ ² (k₁ and k₂ are positive constants) and some regularity hypothesis. Then in the entire domain of definition of the metric W^? one can construct regular oricyclic coordinates (x, y), in which the line element has the form ds² = dx²+ B² (x, y) · dy².     

Evolution of plane-symmetric self-similar space-times containing perfect fluid

Space-times with plane symmetry are considered which also admit a conformal group such that the metric depends on the variablez=x ¹/x ⁰ (homothetic or self-similar solutions). Different regions depending on the space-time character ofz can be discerned in each solution: thet region whenz is timelike and thes-region whenz is spacelike. Exact solutions of the Einstein equations are given for an energy-momentum tensor corresponding to an ideal fluid with equation of statep=(-1)e. Different types of solutions occur depending upon the value of in the equation of state, and these are studied using phase-plane methods. The solutions of considered type in the Minkowski space-time are also given.     

Interacting fields in general relativity theory

The direct interaction of a massless neutral scalar field with an electromagnetic field is investigated with regard for the proper gravitational field. The interaction Lagrangian is chosen in the form L_int=FF, =e–1, where the parameter characterizes the interaction force. Exact static spherically and cylindrically symmetric solutions are obtained. A solution with a finite total field energy is extracted. A comparison is made with the corresponding system in flat space-time. It is concluded that the gravitational field performs a regulatory function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 25–30, September, 1977.The authors are indebted to Yu. S. Vladimirov for valuable comments.     

Instability and reconstruction of a thin fluid film under inversion conditions

The development of mechanical instability of a neutral fluid film (liquid helium or hydrogen) under inversion conditions (it does not lie on a solid substrate but hangs from a ceiling) is discussed. Critical parameters of such an instability and the character of surface reconstruction under the action of van der Waals forces, bubble pressure, and gravitational forces are determined. The interrelation with the well-known Frenkel problem of a drop on a solid substrate is pointed out. An electrostatic mechanism is proposed for the stimulation of instability of a thin helium film. This mechanism is promising for the problem of superfluid helium leakage.     

Conductance of a 2D electron system at low frequencies

Low-frequency features of the conductance of a 2D electron system are considered. It is shown that, in addition to the parameter ωτ (ω is the external frequency and τ is the elastic relaxation time), which occurs in the frequency dependence of the 3D conductivity in the Drude approximation, the 2D conductance contains other dimensionless combinations that involve the external frequency and the 2D conductivity. The notion of the “ mobility” of a 2D system as the quantity governing the deviation of the conductance of the 2D system with ac current from its conductance under stationary conditions is introduced. Experimental data testifying to the presence of the discussed features of the 2D conductance are presented.     

Simple waves in a plasma with anisotropic pressure

Fast and slow simple waves are studied in the framework of the anisotropic magnetohydrodynamics of Chew, Goldberger, and Low [1]. Baranov [2] has constructed fields of integral curves for fast and slow waves and in two special cases has shown that such waves break in the compression section. The possibility of breaking of a slow wave in a rarefaction section was noted by Akhiezer et al. [3]. However, their general relations in simple waves [3] have been shown to be incorrect [2, 4]. In the present paper the nature of the variation of the longitudinal and transverse plasma pressures is determined, and the problem of the breaking of fast and slow waves is completely solved. Conditions under which a slow wave breaks in a rarefaction section are found. A fast wave always breaks in a rarefaction section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 181–183, July–August, 1988.