Scientists and engineers in the theoretical physics sector conduct research in several scientific areas.

**1). Electrically conductive properties of aqueous solutions of electrolytes (academician S. Odinaev, researcher Kh. Idibegzoda). **

**Brief information about the work done.**

Based on the previously obtained analytical expression for the modulus of electroelasticity, when the relaxing fluxes decay exponentially, for a specific model of the solution and explicit expressions for the intermolecular interaction potential and the radial distribution function, depending on the concentration and temperature, a numerical calculation was carried out for an aqueous solution of sodium chloride. Also, based on the analytical expression of the dynamic coefficient of conductivity obtained by the kinetic equation method, with a certain choice of the solution model, the interparticle interaction energy, and the radial distribution function, a numerical calculation was carried out for aqueous solutions of lithium, potassium, and cesium chlorides, depending on concentration, temperature, and frequency.

Based on the ratio of the previously obtained analytical expressions for the dielectric constant and dielectric loss coefficients of electrolyte solutions, the frequency dispersion region of dynamic coefficients and aqueous solutions of LiCl, NaCl, KCl, Ks1, CsCl and KF was studied. Numerical calculations of the friction coefficients, and relaxation times, and, as well as the coefficients in a wide range of changes in density, concentration, temperature, and frequencies are carried out. The theoretically calculated results obtained and the Cole-Cole diagram are in quantitative agreement with the experimental data. Using the method of kinetic equations, analytical expressions for the dynamic coefficient of electrical conductivity are obtained when the relaxing fluxes decay exponentially. The frequency behavior of this coefficient over a wide range of frequencies, as well as asymptotic behavior at low and high frequencies, are investigated. The conductivity coefficient along with molecular parameters contain the potential interaction energy and the radial distribution function. Using a specific model of the solution and explicit expressions for and, we performed numerical calculations of the friction coefficients, relaxation times, and, as well as the conductivity coefficient for aqueous solutions of LiCl, NaCl, KCl, and CsCl depending on concentration, density and temperature. The obtained theoretical data are compared with experimental results, which are in satisfactory agreement.

**2). Research in the field of nonlinear mathematical physics and the theory of condensed matter (academician Kh.Kh.Muminov, leader researcher F.Shokir).**

**Brief information about the work done.**

Based on the numerical simulation of multisoliton solutions of the class of scalar and vector nonlinear Schrödinger equations obtained by the finite-gap integration method, the process of transferring nonlinear excitations in two- and three-level systems is simulated. In particular, the breather dynamics of nonlinear excitations of the multisoliton type, described by the scalar nonlinear Schrödinger equation with the attraction potential in the presence of dissipation and pumping by external fields, was analyzed by numerical simulation methods. The formation of coherent dissipative structures and the appearance of a classical attractor in the phase space of the scalar nonlinear Schrödinger equation with an attractive potential are observed.

As a result of a series of numerical experiments to study the evolution of the two-soliton solution of the nonlinear Schrödinger equation with the repulsion potential, it was shown that the breather dynamics manifests itself when the soliton moves with a nonzero velocity. At the same time, in the presence of dissipation and pumping, even a motionless, static two-soliton solution exhibits breather dynamics. An analysis of the phase portrait of the system shows the formation of a classical attractor, leading to the formation of a stable conservative structure – a long-lived dissipative breather.

The research of spin tunneling in single-molecule Fe8 magnetics is continued by the instanton method using the coherent state of the SU (3) group as a test function. It is shown that the splitting of energy levels occurs due to the presence of a generalized Berry phase in the action, which leads to interference of instanton phase trajectories. It is shown that, taking into account the excitation of quadrupole dynamics, not only the location of the quenching points, but also their number changes. In addition, these quenching points are connected by the number of steps in the hysteresis loop of a given single-molecular magnet. It is shown that when taking into account both dipole and quadrupole excitations in classical energy, the number of steps in the hysteresis loop is equal to the number obtained from experimental data.

A design of a thin-film single-junction solar cell based on ZnSnN2 is proposed. The results of numerical simulations show that the highest efficiency of the proposed design of thin-film solar cells with a p-ZnSnN2 layer thickness of 1100 nm and an n-ZnSnN2 layer thickness of 1320 nm can reach 24.56%, while the open circuit voltage will be 1.69 V and the short-circuit current density – 36.4 mA / cm3.

Studies of a non-Heisenberg ferromagnet with spin value S = 1 with exchange anisotropy in the presence of phonon modulation of spin-quadrupole waves by the method of generalized coherent states of the SU group (3) are continued. A system of equations is obtained for describing the coupled nonlinear spin-quadrupole and sound waves and its solution is obtained. It is shown that the presence of a mechanism of magnetoelastic interactions can lead to an energy exchange between the phonon and spin-quadrupole subsystems.

The simulation of the process of signal transmission between coutrites is carried out. The possibility of transmitting a signal in the form of solitary waves — solitons, which are coupled excited states with internal dynamics in three-level systems, is shown.

Using methods of numerical simulation, we studied the T-symmetry processes of a supersymmetric (2 + 1) -dimensional nonlinear sigma model. Models are obtained that describe the formation of topological vortices during the interaction of localized (topological) disturbances moving along the plane of domain walls and strain waves. When topological vortices collide with certain speeds, they decompose into localized perturbations. In the experiments it was shown that when modeling the final state of this model in reverse time (t ‘= – t) – its initial state is restored. Also, experiments were carried out for evolutionary models of head-on collisions and phased annihilation of topological eddies using the time reversal operation (t ‘= – t). Models are obtained that describe the process of combining radiation waves and the complete formation of the initial state of interacting topological vortices.

Thus, the property of T-invariance of the collision processes of quasiparticles in reverse time in the framework of the n-field is fully confirmed. The results obtained, regardless of the variation of the parameters of the system of interacting quasiparticles and domain walls in all cases confirm the T-invariance property of the equations of the studied n-field.

A study is made of the trajectory of points of a Hilbert complex projective space of systems of generalized spin coherent states in a phase space (Bloch sphere). Based on the Majorana representation, the parameters of 2j unit isospin vectors are obtained for the SU (N) groups (j = 1/2 and j = 1). The Majorana representation of matrices of generalized coherent states is investigated, including the processes of constructing generalized coherent states for spins 1 of group of SU(2) and SU(3) are completely reproduced.

**3). Research in the field of Josephson junction systems (leader researcher I.R.Rakhmonov).**

**Brief information about the work done.**

The current – voltage characteristic of a system of long Josephson junctions taking into account inductive and capacitive coupling is studied. The dependence of the average derivative of the phase difference with respect to time on the magnitude of the base current and the spatiotemporal dependences of the phase difference and magnetic field in each transition are presented. The possibility of branching the current – voltage characteristic in the region of the step of the zero field, which is associated with a different number of fluxons in individual Josephson junctions, is shown. The current – voltage characteristics of the Josephson junctions system are compared with the case of one transition and it is shown that the revealed branching is due to the presence of a connection between the junctions. The intensity of electromagnetic radiation due to the motion of the fluxons is calculated and the influence of the coupling between the transitions on the radiation power is studied.

The results of numerical calculation of the phase dynamics of SQUID (Superconducting Quantum Interference Devices) with topologically trivial and nontrivial barriers are presented. In the calculations, a two-component superconducting current was taken into account, i.e., the current of Cooper pairs (2pi periodic) and the current of Majorana fermions (4pi periodic). The dependence of the return current on the magnetic field is presented. It is demonstrated that in the case of a two-component superconducting current, the periodicity of the dependence of the return current on the magnetic field shifts by an amount determining the ratio of the currents of Cooper pairs and Majorana fermions.

The Josephson current between two one-dimensional nanowires with induced superconducting s-wave pairing and separated by a dielectric barrier in the presence of a spin-orbit interaction and a Zeeman field is calculated. A generalized method for calculating the energy of the Andreev state is formulated, which allows one to obtain analytical expressions for the energy of these states in some asymptotic cases. It was found that in the absence of a magnetic field, the energy gap between the Andreev states decreases with an increase in the Rashba spinorbital interaction constant. In the absence of Rashba’s spin-orbit interaction, the Andreev states depend on the magnetic field and exhibit an oscillating character with a change in the magnetic field, leading to the magnetic Josephson effect. Analytical expressions for constant Josephson current are also shown. The dynamics of the Josephson transition taking into account spin-orbit interaction is also studied. The dynamics of a superconductor-ferromagnet-superconductor type structure was studied, which is also called the Phi-0 transition in the literature. The main area of research was associated with the identification of the possibility of a complete revolution of the magnetic moment in the Phi-0 transition in various ways. The possibility of flipping the magnetic moment through a rectangular current pulse is shown. The dynamics of a superconducting quantum interferometer containing one Phi-0 junction is studied. It is shown that in such a quantum interferometer, it is possible to realize a complete flip of the magnetic moment under the influence of an external magnetic field pulse. Various protocols have been proposed for a complete flip of the magnetic moment with a variation of the Phi-0 transition parameters, pulse parameters of the electric current, and parameters of the magnetic system. The possibility of the experimental implementation of a complete revolution of the magnetic moment is analyzed.

**4). Research in the field of quantum-mechanical calculations of the dynamics of the molecular complex (leader researcher A.G.Jabarov).**

**Brief information about the work done.**

Quantum-mechanical calculations of obtaining a stable configuration of a molecular complex composed of molecules of silicon dioxide (SiO2) and ethyleneol (C2H3OH) were performed. The calculated complex was supposed to be used as a model of the electron trap center in polyethylene filled with silicon dioxide. The equilibrium geometry, energy states, binding energy, IR, Raman scattering, and UV spectra of a neutral and charged molecular complex are calculated.

The analysis of the results led to the following conclusions:

- all calculated molecular structures have an energy minimum – they are stable formations;
- the addition of an extra electron to the molecular complex 1 leads to a change in the structure of the complex (change in valence angles, lengths of valence bonds);
- charging the complex leads to a change in the distribution of energy density and charge;
- the total energy decreases (by almost 2.5 eV, which makes the charged state more favorable), the dipole moment increases;
- a small binding energy (0.25 eV) of the molecular complex during the formation of a hydrogen bond between the hydroxyl group of C14H27OH and one of the silicon dioxide oxygen increases significantly when an excess electron is attached (by more than 2.5 eV, the O – H bond length increases by about 5%) ;
- The orbital filled with an excess electron is mainly localized on silicon dioxide, although the states of the nearest C14H27OH atoms to SiO2 make a small contribution.
- An increase in the dipole moment and an increase in the hydrogen bond upon the addition of an extra electron by the complex leads to two noticeable effects, first, an increase in the intensity of the absorption bands of IR radiation associated with vibrations of the groups involved in the formation of the hydrogen bond, and secondly, their significant shift into the long wavelength region. This is clearly seen in the example of the 3681 cm – 1 band, which in the charged complex shifts to the 3140 cm – 1 band, increasing significantly in intensity. UV spectra after the addition of an excess electron are shifted to the long-wavelength side. The first calculated optical transition from the long-wave side corresponds to an energy of 3.5 eV (4 eV in the neutral complex). The binding energy of the charged complex is 2.5369 + 0.2504 = 2.7873 eV. The activation energy of a homo charge trapped in deep traps in samples from a polyethylene – silicon dioxide (aerosil) composite is greater than 1 and less than 2 eV. Taking into account the activation mechanism of homo-charge relaxation, as well as the fact that the homo-charge is released from traps when the samples are heated at temperatures corresponding to the thawing of αc-molecular mobility, the process of homo-charge relaxation occurs mainly as a result of the destruction of molecular complexes due to intense thermal motion of macromolecules.

The calculation results are reflected in the report “On the nature of deep traps in electrets made of silicon dioxide-filled polyethylene” at the VI International Conference “Modern Problems of Physics”. PhTI, Dushanbe, 2018.

Spectroscopic studies of glasses of various compositions (Bi2O3 – Na2B4O7) obtained with various technological factors were carried out. The transmission spectra (at normal incidence of the light beam) and reflection (at different angles of incidence of the light beam) from the near UV region to the near IR region (194 – 1006 nm) were measured. The dependences of the absorption coefficient k and the reflection coefficient R on the wavelength are calculated. Studies were carried out on samples that differed in composition and technological features of their manufacture. Using a Shimadzu Fourier spectrophotometer, IR spectra were obtained for some glass samples of the composition (xBi2O3 – (x-1) Na2B4O7). Spectra were obtained for the following wave number intervals: 350-4000 (survey spectrum); 350-1200; 1200-2500 and 2500-4000 cm-1.

**5). Research in the field of density functional theory (leader researchers K.S.Mekhrdod, F.Shokir, Kh.R.Rakhmonov, engineer Sh.Yu.Aminjonov)**

**Brief information about the work done. **

One of our major scientific work is the theoretical calculations (quantum) within a density functional theory (DFT) to investigate the geometric and electronic structures of various kinds of functional materials. These scientific works have been mostly done under collaborations with the research group of Prof. Tomoyuki Yamamoto, Waseda University, Tokyo, Japan and other foreign scientific institutions.

Most of the current phosphor materials for the industrial use are synthesized using a doping technique, i.e., incorporation of a trace element as an emitting centre in the matrix materials. Doping of rare-earth and 3d transition elements can produce the efficient phosphor materials. For the red-emitting phosphors, rare-earth ions, such as Pr and Eu, have been used. Because of the view point of stability for long term use, oxides are selected as matrix materials. Among such doped phosphors, rare-earth doping in the host materials with the wide band-gap such as oxides and fluorides is quite efficient. However, due to the limitation of the rare-earth elements in the earth, rare-earth free phosphor materials have been strongly demanding, and therefore such materials have been extensively investigated these years. On the other hand, 3d transition elements are also good dopants for such purpose, in which Mn4+ is the best candidate for the red-emitting rare-earth free phosphor in the coming generation.

In order to design new phosphors doped with Mn ions, it is essential to know the local environment of the doped Mn ions in an atomic scale and the electronic structures of Mn-doped materials. In order to enhance luminescence efficiency of the Mn doped phosphors, it is also very important to understand the electronic structures of the materials. For such purpose, the first principles calculations within a density functional theory are quite useful. The various kinds of oxides with perovskite structures doped with Mn ions were systematically have been examined by the state of the art first principles DFT calculations. Some calculated results were compared with the experimental results, such as X-ray diffraction, diffuse reflectance spectra, and photoluminescence spectra, etc., provided by the research group of Prof. Yamamoto.