On the variational representation of solutions to some quasilinear equations and systems of hyperbolic type on the basis of potential theory

We demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type. The corresponding variational principle is constructed using the theory of equilibrium of a potential in an external field. Dedicated to the memory of B. M. Levitan Supported by RFBR grants Nos. 05-01-00522 and NSh-1551.2003.1, by Program No. 1 of the Branch of Mathematics, Russian Academy of Sciences, and by INTAS project No. 03-51-6637.     

Observation of the nonlinear effect in relativistic Thomson scattering of electron and laser beams

Thomson scattering of high-power laser and electron beams is a good test of electrodynamics in the high-field region. We demonstrated production of high-intensity X-rays in the head-on collision of a CO₂ laser and 60-MeV electron beams at Brookhaven National Laboratory, Accelerator Test Facility. The energy of an X-ray photon was limited at 6.5 keV in the linear (lowest order) Thomson scattering, but the nonlinear (higher order) process produces higher energy X-rays. We measured the angular distribution of the high-energy X-rays and confirmed that it agrees with theoretical predictions.     

Direct C–H arylation of 3-methoxythiophene catalyzed by Pd. Application to a more efficient synthesis of π-alkoxy-oligothiophene derivatives

The direct regioselective C–H arylation of 3-methoxythiophene and 3,4-ethylenedioxythiophene (EDOT) was performed successfully under ‘Heck-type’ experimental conditions. This novel synthetic methodology has been used to prepare in a more simple way a series of oligothiophenes interesting for the electronic industry to build new synthetic organic materials.     

Stereocontrolled spirocyclic bislactams derived from pyroglutamic acid

The synthesis of rigid spirocyclic bislactams derived from pyroglutamic acid has been established.     

Stability of one-phase equilibrium states of a two-phase elastic medium with zero surface tension coefficient. One-dimensional case

All the equilibrium states of a one-dimensional variational phase-transition problem are explicitly found. The temperature-dependence of the stability of one-phase equilibrium states is studied. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 32, 2006, pp. 3–19.     

Time evolution of non-Hermitian quantum systems and generalized master equations

We inquire into the time evolution of quantum systems associated with pseudo-or quasi-Hermitian Hamiltonians. We obtain, in the pseudo-Hermitian case, a generalized Liouville-von Neumann equation for closed systems. We show that quantum systems with quasi-Hermitian Hamiltonians admit the proper interpretation in terms of open quantum system and derive a generalized Lindblad-Kossakowski equation. Finally, we extend such formalism to the study of decaying systems. Partially supported by PRIN “Sintesi”.     

Spatial structure of acousto-optic phase matching in uniaxial crystals

The spatial structure of Bragg angles and the transfer functions of an acousto-optic cell are calculated for the cases of isotropic and anisotropic light diffraction in a uniaxial crystal. Their change with ultrasound frequency is traced. The possibility of image processing by acousto-optic filtration of the spatial spectrum of an image is considered. The results of experimental visualization of the transfer functions of a calcium molybdate crystal cell are presented.     

Damped waves in a cylinder shadow

The sound field in the region of a deep shadow behind an impedance cylinder is analyzed for the case of plane wave diffraction by the cylinder surface. The main part of the field is represented by a cylindrical wave that has a complex index determined from the boundary conditions and decays with the angular coordinate. An expression for the amplitude of this wave is determined by extracting it from the total field formed in the shadow region. It is demonstrated that this wave approximates more closely the total field behind the cylinder, as compared to the least damped wave in the field representation obtained on the basis of the Watson transform. A way to improve the sound barriers is indicated.