Synchronization between the internal dynamics of the superconducting phase in a Josephson junction (JJ) and an external ac signal is a fundamental physical phenomenon, manifesting as constant-voltage Shapiro steps in the current-voltage characteristic. Mathematically, this phase-locking effect is captured by the Resistively Shunted Junction (RSJ) model, an important example of a nonlinear dynamical system. The standard RSJ model considers an overdamped JJ with a sinusoidal (single-harmonic) current-phase relation (CPR) in the current-driven regime with a monochromatic ac component. While this model predicts only integer Shapiro steps, the inclusion of higher Josephson harmonics is known to generate fractional Shapiro steps. In this paper, we show that only two Josephson harmonics in the CPR are sufficient to produce all possible fractional Shapiro steps within the RSJ framework. Using perturbative methods, we analyze amplitudes of these fractional steps. Furthermore, by introducing a phase shift between the two Josephson harmonics, we reveal an asymmetry between positive and negative fractional steps — a signature of the Josephson diode effect.

We develop a theoretical framework, using the slow variables method in the RSJ model, and perform numerical simulations to explain the peculiarities of the Josephson diode effect observed in an asymmetric SQUID with a superconducting nanobridge and an SNS junction in recent experiments by Vasiliy Stolyarov’s group at MIPT. For this case we predict the new (amplitude) mechanism of the diode effect which originates from the presence of a superconducting nanobridge exhibiting a multivalued current-phase relation (CPR). Theoretically, this mechanism manifests itself in the dependence of the amplitude of the first Josephson harmonic in the CPR of the SQUID on the current direction. We demonstrate that this mechanism accounts for the experimentally observed pronounced asymmetry of the Shapiro steps, which coexists with a relatively small asymmetry in the critical current. Additionally, we investigate how the diode efficiency (quantified via the Shapiro steps asymmetry) depends on both the magnetic flux through the superconducting loop and the power of the external microwave irradiation. Our theoretical predictions are found to be in good agreement with the experimental observations. This talk is based on the article Phys. Rev. B 112, 144504 (2025).

The Novikov problem on the level lines of quasiperiodic potentials on a plane with additional (quasi-crystalline) symmetry is considered. For potentials of this type, the possibility of the emergence of open level lines at only a single energy level is shown in the general case. Possible estimates of the growth rate of closed level lines as the energy approaches the percolation threshold are also discussed.

We study triggered by an a.c. external electric field weakly nonlinear stages of flexoelectric instability in nematic liquid crystals. The instability occurs at a finite wave vector. We analyze behavior on time scales much larger than the period of the external electric field. We focus on the case where the increment of the most-unstable mode has an imaginary part, so-called Hopf bifurcation. The existence of such regime was established in our previous work [E.S. Pikina, A.R. Muratov, E.I.Kats, V.V. Lebedev, Dynamic flexoelectric instabilities in nematic liquid crystals, Phys. Rev. E, 110, 024701 (2024)]. Then above the instability threshold a variety patterns of nematic director distortions could appear including standing and travelling structures. Our numerical simulations based on the full nonlinear electro-nematodynamics system of equations. We found that the stable dynamic pattern in the vicinity of the Hopf bifurcation travelling oblique rolls of the nematic director distortions. The establishment of this regime occurs abnormally slowly, which is determined not only by the critical Landau-like slowdown of dynamics, but also by the presence of a long-lived intermediate unstable but long-lived dynamic patterns oscillating in time (standing but not traveleling rolls). Depending on liquid crystal material parameters, the bifurcation corresponding to the formation of the travelling oblique rolls, can be soft (i.e.continues, ‘’critical’’ or close to ‘’tricritical’’ one), or hard (discontinues).

Мы развиваем феноменологическую теорию, описывающую взаимодействие сверхпроводящего конденсата с Бесселевым пучком структурированного (закрученного) света, характеризуемого ненулевым орбитальным угловым моментом $m$. Стартуя с нестационарной теории Гинзбурга-Ландау с комплексной константой релаксации, мы вычисляем пространственные распределения стационарных (dc) фотоиндуцированных токов и магнитных полей, а также отклик на второй гармонике. Мы показали, что фототоки и магнитные поля определяются как поляризацией света, так и его орбитальным моментом $m$. Мы проанализировали две различные геометрии: сверхпроводящего полупространства и тонкой плёнки. В конце мы обсуждаем возможные экспериментальные методики для измерения сверхпроводящих фототоков и соответствующих магнитных полей.

The unitarity of the 4D lattice theory of gravity in the case of the Minkowski signature is proved. The proof is valid only for lattices that conserve the number of degrees of freedom during time evolution. The Euclidean signature and the Minkowski signature are related by the deformation of the integration contours of dynamic variables in a discrete lattice functional integral. It is important that the result is obtained directly on the lattice. Since the studied lattice theory of gravity in the long-wave limit transforms into the well-known Einstein-Cartan-Palatini theory, the obtained result means that this lattice theory of gravity has the right to be considered as a discrete regularization of the generally accepted continuous physical theory of gravity.