Various phenomena in nano-systems, based on graphene, topological insulators and semiconductor heterostructures, are studied. In particular, influence of Coulomb interaction on quantum capacitance and compressibility of a gas of massless Dirac electrons in graphene is analyzed. Highly sensitive method of intracavity spectroscopy, based on usage of spaser – quantum generator of surface plasmon polaritons – is proposed. The formation of the roton-maxon excitation spectrum and the roton instability effect for a weaklycorrelated Bose gas of dipolar excitons in a semiconductor layer are predicted.

We performed numerical simulation of systems and phases with symmetry groups associated with liquid crystalline order. We have derived the equations for 1D textures of biaxial nematics. To describe the conformations of the above textures we introduce the curvature vector that corresponds to a skew-symmetric matrix describing infinitesimal rotations of the order parameter while moving along the axis of the texture. The construction is helpful for the analysis of data provided by the numerical modeling. The existence of quasi-periodic solutions has been shown.

For planar Penning nanotrap in the primary and secondary resonance regime we computed the effective Hamiltonian with one degree of freedom over the symmetry algebra with quadratic commutation relations. We obtained explicit formulas for its equilibrium points and their dependence from control parameters of the trap.

We show that the diagonalization problem for multimode squeezings can be reduced to the Takagi factorization of symmetric matrices, and the diagonalization procedure is numerically unstable for parameterized systems of symmetric matrices at the points, where the multiplicity is not preserved. The normal factorization of the squeezing whose Hamiltonian contains the particle number operator requires selecting a proper branch of the square root of Jacobians in coefficients. The Wolfram Mathematica environment tools were developed and tested for the factorization of symmetric matrices, for the normal factorization of multimode squeezings, for calculating the inner products of generalized multimode squeezings.

The role of quantum entanglement in the transmission of information by quantum communication channels is investigated. Bosonic Gaussian classical-quantum (c-q) channels were introduced and studied. For single-mode nondegenerate bosonic Gaussian channels the optimality of the coherent input states is proved. It made possible to establish that the classical capacity of such channels is achieved by Gaussian encoding, resulting in explicit expressions of the fundamental limits for the information transfer for the most commonly used models of single-mode Gaussian quantum channels.

The model of heat transfer in the process of electron emission from cathodes of small sizes is investigated by taking into account the complicated geometry of the cathode tip. A method is developed for solving the heat transfer equations and the Allen-Cahn resolving problem which are related to the singularity of the Laplace operator coefficients at the origin of the spherical coordinate system. The construction of a weak asymptotic solution of the phase field system is developed for the melting-crystallization problem in nanocathodes.

An asymptotic solution of the problem of incompressible viscous liquid flow in a two-dimensional channel with small imperfections on the walls is obtained for large Reynolds numbers and in the case of an axially symmetric tube. It is shown that the boundary layer has a double-deck structure, i.e., a thin near-wall boundary layer and a “thick” boundary layer. An algorithm for solving the equations describing the flow in the thin boundary layer is developed and numerical simulation of this flow is performed. The influence of the imperfection amplitude and the channel width on the flow character in the thin boundary layer is studied.

A Rayleigh-type equation describing the oscillations in the “thick” boundary layer was studied in the case of flow past a plate. The existence and uniqueness of its stationary solution are proved, as well as the solution stability at large times under natural assumptions on the character of the boundary oscillations. The results of numerical simulation which show the difference between the classical Prandtl-Blasius flow for a flat plate and the flow in the Prandtl boundary layer for a rough plate are given.

The entropy of the system of dust particles in plasma is investigated.The methods of configurational entropy calculation and other methods of entropy estimations are presented. A model of dust particles in a gas discharge plasma with account of fluctuations in the charge of the particles, the dependence of the charge on the distance from the electrode to the other, and dust particles, also the features of the sheath discharge are created. Dusty plasma is simulated by molecular-dynamics method.

In the framework of the electronic density functional method the values of the reflectivity coefficient for shock compressed xenon plasma has been calculated. The results are compared with experimental measurements.

New statistical models have been developed to estimate a phase point transition in a minimum planar matching problem on a random graph. Particularly, a model of non-integer alphabet in random RNA-type molecules, described in terms of planar matching structure was studied in details.

A new concept of molecular machine designing based on the crumpled (fractal) polymer globules is proposed. The prototype of a globular structure that can convert local energy excitation in to directed nano-scale motions is found using the computer simulation methods. In addition, the conditions for self-organization of the molecular machines by means of collapsing of polymer chains in to crumpled conformations is specified as well.