Inverse medium problem for a singular contrast

We consider an inverse medium problem in two- and three-dimensional cases. Namely, we investigate the problem of reconstruction of unknown compactly supported refractive index (contrast) from L-2 with a fixed positive wave number. The proof is based on the new estimates for the Green-Faddeev function in L-infinity space. The main goal of this work is to prove a uniqueness result in the two- and three-dimensional cases and to discuss some possible constructive methods for solving the problem. Finally, we present some numerical examples to demonstrate the results in two dimensions. Published under license by AIP Publishing.Peer reviewe

Topological Coherent Modes for Nonlinear Schr\"odinger Equation

Nonlinear Schr\"odinger equation, complemented by a confining potential, possesses a discrete set of stationary solutions. These are called coherent modes, since the nonlinear Schr\"odinger equation describes coherent states. Such modes are also named topological because the solutions corresponding to different spectral levels have principally different spatial dependences. The theory of resonant excitation of these topological coherent modes is presented. The method of multiscale averaging is employed in deriving the evolution equations for resonant guiding centers. A rigorous qualitative analysis for these nonlinear differential equations is given. Temporal behaviour of fractional populations is illustrated by numerical solutions.Comment: 14 pages, Latex, no figure

Wave packet evolution approach to ionization of hydrogen molecular ion by fast electrons

The multiply differential cross section of the ionization of hydrogen molecular ion by fast electron impact is calculated by a direct approach, which involves the reduction of the initial 6D Schr\"{o}dinger equation to a 3D evolution problem followed by the modeling of the wave packet dynamics. This approach avoids the use of stationary Coulomb two-centre functions of the continuous spectrum of the ejected electron which demands cumbersome calculations. The results obtained, after verification of the procedure in the case atomic hydrogen, reveal interesting mechanisms in the case of small scattering angles.Comment: 7 pages, 8 Postscript figure

Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page

Influence of the Nanostructures on the Surface and Bulk Physical Properties of Materials

Fullerenes, nanotubes, quantum dots are considered as effective sensitizers to modify both the optical, nonlinear optical features, dynamic and polarization characteristics, as well as mechanical and spectral properties of the organic and inorganic materials. The correlation between photorefractivity and photoconductivity was supported and the relation between charge carrier mobility of pure conjugated structures and nanoobjects-doped ones has been revealed. An increase of transmission of nanostructured polarization films was observed. An extension of the nanocomposites applications area is considered

Symmetry Influence on the Rotation of Molecules in Crystals

© 2017 American Chemical Society. It was shown that rotational mobility of molecules in crystals is affected by the symmetry of their surroundings. A hypothesis was proposed for the discovered correlation. Three cases are possible for the location of the molecules with respect to the crystallographic symmetry elements: I - the location in a general position; II - the location in special positions without symmetry disordering; III - the location in special positions with symmetry disordering. According to the experimental data, the rotation barrier heights at the location of the molecules in cases I and III are lower than in case II. This fact is explained by the amplitude and p hase shifts of the rotational energy profiles of two parts of the molecule in case I and by increasing the number of minima on the rotation barrier profile at disordering the molecules by symmetry in case III. The way is proposed for lowering the rotational barrier of molecules in crystals