In the talk, I will discuss the examples of integrable differential-difference equations of order 2 with respect to the discrete variable constructed in [1]. Over the past few years, some particular classification results have been obtained for such type equations, but their general description remains a very difficult open problem. The considered examples are related with continuous symmetries of discrete equations on a triangular lattice. I demonstrate that their linear combination can be written as a scalar chain of order 2, under restriction at one of the lattice axes. Although this construction is rather special, helps to significantly expand the list of known examples. [1] V.E. Adler. Integrable seven-point discrete equations and second-order evolution chains. Theor. Math. Phys. 195:2 (2018) 513-528.

Most studies of the localized edge (EM) and defect (DM) modes in cholesteric liquid crystals (CLC) are related to the localized modes in a collinear geometry, i.e. for the case of light propagation along the spiral axis. Much less attention was paid to the localized modes in CLC for a non-collinear geometry. It is due to the fact that all photonic effects in CLC are most pronounced just for the collinear geometry and also partially due to the fact that a simple exact analytic solution of the Maxwell equations is known for the collinear geometry, whereas for a non-collinear geometry there is no exact analytic solution of the Maxwell equations and a theoretical description of the experimental data becomes more complicated. It is why in papers related to the localized modes in CLC for a non-collinear geometry and observing phenomena similar to the case of a collinear geometry their interpretation is not so clear. Problems related to the localized modes for a non-collinear geometry are studied here in the two wave dynamic diffraction theory approximation. The dispersion equation for non-collinear localized edge modes (called conical modes (CEM)) is found and analytically solved for the case of thick layers and for this case found the lasing threshold and the conditions of the anomalously strong absorption effect. Shown that qualitatively CEMs are very similar to the EMs, however differing by their polarization properties (the CEM eigen polarizations are elliptical one depending on the degree of CEM deviation from the collinear geometry in contrast to the circular eigen polarizations in the EM case). What is concerned of the CEM quantitative values of the parameters they are “worth” than for the corresponding ones for EM. The CEM lasing threshold is higher than the one for EM and etc. Performed theoretical studies of possible conversion of EMs into CEMs showed that it can be due to the EM reflection at dielectric boundaries at the conditions of a high pumping wave focusing. Known experimental results on the CEM are discussed and optimal conditions for CEM observations are formulated.
1. V.A. Belyakov, S.V. Semenov, Localized conical edge modes of higher orders in photonic liquid crystals, Crystals, 9(10), 542 (2019);
2. V.A. Belyakov, Localized Conical Edge Modes in Optics of Spiral Media (First Diffraction Order), Crystals, 9(12), 674 (2019).

One of the most exciting phenomena observed in crystalline disordered membranes, including a suspended graphene, is rippling, i.e. a formation of static flexural deformations. Despite an active research, it still remains unclear whether the rippled phase exists in the thermodynamic limit, or it is destroyed by thermal fluctuations. We demonstrate that a sufficiently strong short-range disorder stabilizes ripples, whereas in the case of a weak disorder the thermal flexural fluctuations dominate in the thermodynamic limit. The phase diagram of the disordered suspended graphene contains two separatrices: the crumpling transition line dividing the flat and crumpled phases and the rippling transition line demarking the rippled and clean phases. At the intersection of the separatrices there is the unstable, multicritical point which splits up all four phases. Most remarkably, rippled and clean flat phases are described by a single stable fixed point which belongs to the rippling transition line.

Investigation of Josephson effect, current flow in narrow superconducting stripes, dynamical states in superconductors lead to the necessity to deal with an important phenomenon: phase slip events. The study of the distribution of zeros for Riemann's Zeta function also requires an analisis of the same phenomenon.
It was found that, in addition to trivial zeros in points ($ z = -2N, N = 1, 2, ... $, natural numbers), the Riemann’s zeta function $\zeta(z)$ has zeros only on the line {$z = 1/2 + i t_0$, $t_0$ is real}. All zeros are numerated, and for each number, N, the positions of the non-overlap intervals with one zero inside are found. The simple equation for the determination of centers of intervals is obtained. The analytical function $\eta(z)$), leading to the possibility fix the zeros of the zeta function $\zeta(z)$, was estimated. To perform the analysis, the well-known phenomenon, phase-slip events, is used. This phenomenon is the key ingredient for the investigation of dynamical processes in solid-state physics, for example, if we are trying to solve the TDGLE (time-dependent Ginzburg-Landau equation).
J. Supercond. Novel Magn., 32(11), 3363-3368 (2019)