# In Print

## Zeros of Riemann’s Zeta Functions in the Line z=1/2+it_{0}

25 September in 11:30

Yu.N. Ovchinnikov

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)

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)

## Perturbed Sachdev-Ye-Kitaev model: a polaron in the hyperbolic plane.

25 September in 11:30

__A. V. Lunkin__, A. Yu. Kitaev, and M. V. Feigel'man

We study the SYK model with a weak quadratic perturbation, beyond the simplest
perturbative limit considered previously. For intermediate values of the perturbation strength, fluctuations of the Schwarzian mode are suppressed, and the mean-field
solution remains valid beyond the expected interval. Out-of-time-order correlation function displays at short time intervals exponential growth with maximal Lyapunov exponent 2\piT, but its prefactor scales as T at low temperatures

## Zerkal’nye pary orbifoldov Kvintiki

18 September in 11:30 (short)

__B. Eremin__, A. Belavin

We compare two constructions of mirror pairs of Calabi-Yau manifolds using the example of Quintic orbifolds. We consider the quotient Q/H of the original Calabi-Yau by the subgroup H of the maximal allowed group. Then the mirror variety is defined by an additional subgroup of HT as Q/HT. We compare this result with the mirror obtained in the Batyrev approach and show their equivalence.

## Localized conical edge modes and laser emission in photonic liquid crystals

11 September in 11:30 (short)

V.A. Belyakov

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).

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).

## Non-Abelian evolution systems with conservation laws and symmetries

11 September in 11:30

V.V. Sokolov

We find noncommutative analogs for well-known integrable polynomial systems of differential equations in two unknowns.

1. V. Sokolov, T. Wolf, Non-commutative generalization of integrable quadratic ODE systems, Lett. Math. Phys., 110(3), 533-553 (2020)

2. V.E. Adler, V.V. Sokolov, Non-Abelian evolution systems with conservation laws, arXiv:2008.09174

1. V. Sokolov, T. Wolf, Non-commutative generalization of integrable quadratic ODE systems, Lett. Math. Phys., 110(3), 533-553 (2020)

2. V.E. Adler, V.V. Sokolov, Non-Abelian evolution systems with conservation laws, arXiv:2008.09174

## Maxwell equations in curved space-time: non-vanishing magnetic field in pure electrostatic systems

4 September in 11:30

N.N. Nikolaev and

__S.N. Vergeles__
Solutions of the Maxwell equations for electrostatic systems with manifestly vanishing electric currents in the curved space-time for stationary metrics are shown to exhibit a non-vanishing magnetic field of pure geometric origin.
In contrast to the conventional magnetic field of the Earth it can not be screened away by a magnetic shielding. As an example of practical significance we treat electrostatic systems at rest on the rotating Earth and derive the relevant geometric magnetic field.
We comment on its impact on the ultimate precision searches of the electric dipole moments of ultracold neutrons and of protons in all electric storage rings.

## Lyapunov exponent for Whitney's problem with random drive

4 September in 11:30

__Nikolai A. Stepanov__, Mikhail A. Skvortsov

We consider the statistical properties of a non-falling trajectory in the Whitney problem of an inverted pendulum excited by an external force. In the case when the external force is white noise, we recently found the instantaneous distribution function of the pendulum angle and velocity over an infinite time interval using a transfer-matrix analysis of the supersymmetric field theory. Here, we generalize our approach to the case of finite time intervals and multipoint correlation functions. Using the developed formalism, we calculate the Lyapunov exponent, which determines the decay rate of correlations on a non-falling trajectory.

## Inverted pendulum driven by a horizontal random force: statistics of the non-falling trajectory and supersymmetry

5 June in 11:30

__Nikolay Stepanov__, Mikhail Skvortsov

We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction. Considered at the entire time axis, the problem admits a unique solution which always remains in the upper half plane. We develop a new technique for treating statistical properties of this unique non-falling trajectory.
In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistics of the non-falling trajectory is expressed in terms of the zero mode of a corresponding transfer-matrix Hamiltonian.
The emerging mathematical structure is similar to that of the Fokker-Planck equation, but it is rather written for the «square root» of the distribution function.

We derive the specific boundary conditions that correspond to the non-falling trajectory. Our results for the distribution function of the angle and its velocity at the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of very strong noise, an exact analytical solution is obtained.

We derive the specific boundary conditions that correspond to the non-falling trajectory. Our results for the distribution function of the angle and its velocity at the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of very strong noise, an exact analytical solution is obtained.