We illustrate some new results and comment on perspectives of a recent research line, focused on the stability of stationary states of nonlinear NLS with point interactions. We describe in detail the case of a “𝛿′” interaction, that provides a rich model endowed with a pitchfork bifurcation with symmetry breaking in the family of ground states. Finally, we give a direct proof of the stability of the ground states in the cases of a subcritical and critical (in the sense of the blow-up) nonlinearity power. 

The Casimir force between two identical bodies, although highly dependent on their geometry and structure of boundaries, is always attractive. However, this force can become repulsive if the nature of the two boundaries is different. We analyze from a global perspective the analytic properties of the Casimir energy function in the space of the consistent boundary conditions ℳ_F for a massless scalar field confined between two homogeneous parallel plates. The analysis allow us to completely characterize the boundary conditions which give rise to attractive and repulsive Casimir forces. In the interface between both regimes there is a very interesting family of boundary conditions which do not generate any type of Casimir force. We also find Casimirless boundary conditions which are invariant under the renormalization group flow. The conformal invariant boundary conditions which do not generate a Casimir force have not yet been exploited in string theory but open new interesting possibilities.

Bloch-waves in 1D periodic lattices are typically constructed based on the transfer-matrix approach, with a complete system of solutions of the Cauchy problem on a period. This approach fails for the multi-dimensional Schr¨odinger equations on periodic lattices, because the Cauchy problem is ill-posed for the associated elliptic partial differential equations. In our previous work [8] we suggested a different procedure for the calculation of the Bloch functions for the 2D Schr¨odinger equation based on the Dirichlet-to-Neumann map substituted for the transfer -matrix. In this paper we suggest a method of calculation of the dispersion function and Bloch waves of quasi-2D periodic lattices, in particular of a quasi-2D sandwich, based on construction of a fitted solvable model. 

Consider the quasi-static irreversible evolution of a connected network, which minimizes the average distance functional. We look for conditions forcing a bifurcation, thus changing the topology. We would give here a sufficient conditions. Then we will give an explicit example of sets satisfying the bifurcation condition, and analyze this special case. Proofs given here will be somewhat sketchy, and this work is based on [9], in which more details can be found.

In this article, we consider two initial-boundary value problems with nonlocal conditions. The main goal is to show the method which allows to prove solvability of a nonlocal problem with integral conditions of the first kind. This method is based on equivalence of a nonlocal problem with integral conditions of the first kind and nonlocal problem with integral conditions of the second kind in special form. Existence and uniqueness of generalized solutions to both problems are proved.

We give a review of recent developments on the study of threshold eigenfunctions and threshold resonances of magnetic Dirac operators and Pauli operators. Emphasis is placed on a proof of the non-existence of threshold resonances of the magnetic Dirac operators in a concise manner.

We study the director orientation and variation of the layered structure of the chevron ferroelectric smectic C* in a bounded cell in presence of an external electric field. The spatial distribution of the director was calculated in systems possessing the orientational bistability. We analyze the transition between two equilibrium configurations in weak electric fields when the magnitude and direction of the field are changed. The transition has the form of a hysteresis loop. The value of the threshold field leading to the reorientation of the director was estimated. The variation of the layered structure occurs in strong fields when the contribution of the quadratic term of the field to the free energy is essential. In this case the spatial and orientational structure of smectic C* was obtained for different ratios between the principal values of the permittivity tensor. It was shown that in some cases the increase of the external field should lead to the transition from the chevron to the "bookshelf" structure.

On the basis of molecular dynamics method the algorithm for the first time enables to simulate a plane flow in nanochannels with a pressure drop is proposed. Interaction between molecules of the fluid is simulated by the potential of hard spheres or the Lennard-Jones potential. The properties of nanoflows are studied. It is shown that the structure of fluids in nanochannels differs significantly from its structure in the bulk. Data on the dependence of friction coefficient on the Knudsen and Reynolds numbers are presented. It is established that the pressure drop depends strongly on the accommodation coefficients (for the fluid of hard spheres) or the parameters of the interaction of fluid molecules with molecules of the wall (for the Lennard-Jones fluid).

Results of numerical simulation of formation and growth of particles of a titan dioxide in a working zone of the plasma-chemical reactor are presented. Influence of heterogeneous reaction and nucleation of the particles is studied. Parametrical calculations of formation and growth of particles of a titan dioxide depending on rate and velocity injection of the gas-phase of TiCl4 are carried out.

Values of the critical surface tension of the liquid (ethanol solution containing different amounts of water) on the boundary with the solid body (fraction of sandy soil with different dispersion) were calculated, using the method of Zisman. Greatly increased supply of surface free energy and the strengthening of inter-particle interaction with increasing of dispersion were established. Based on this research, were proposed to use a constant Hamaker and critical surface tension as a criteria for the formation and characteristics of the surface state of the dispersed material. This will allow us to estimate the strength of the relationship of composites based on nanostructures.

It is shown that chemical prehistory of chromium hydroxide has an influence on dehydration and crystallization of Cr₂O₃-H₂O system under thermal and hydrothermal treatment. Parameters of hydrothermal treatment allowing one to obtain nanocrystals of Cr₂O₃ are determined.