The strong rotation makes the turbulent flow quasi-two-dimensional, that leads to the transfer of energy on a large scale. Recent numerical simulations show that under certain conditions, energy accumulates on the largest scales of the system, forming coherent vortex structures known as condensates. We have carried out an analytical description of the interaction of a strong condensate with weak small-scale turbulent pulsations and obtained an equation that makes it possible to determine the radial profile of the azimuthal velocity of a coherent vortex. With a fast external rotation, the velocity profiles of cyclones and anticyclones are identical to each other and are well described by a linear-logarithmic dependence. As the external rotation decreases, this symmetry disappears: the maximum velocity in cyclones is higher, and the position of the maximum is closer to the axis of the vortex in comparison with anticyclones. In addition, our analysis shows that the size of the anticyclone cannot exceed a certain critical value, which depends on the Rossby and Reynolds numbers. The maximum size of cyclones is limited only by the size of the system under the same conditions. Next, we took into account the boundary effects. For typical experimental conditions, the profile of the condensate velocity at sufficiently large distances from the vortex axis is determined by the Ekman linear friction associated with the no-slip conditions at the lower and upper flow boundaries. At the distances, the azimuthal velocity of a coherent vortex does not depend on the distance to the center of the vortex and is determined by the energy balance between the pumping power and friction against the boundaries. We investigate the structure of a coherent vortex in this case and compare the results with the profile of the condensate velocity in two-dimensional systems.
[1] Vladimir Parfenyev and Sergey Vergeles. Influence of Ekman friction on the velocity profile of a coherent vortex in a three-dimensional rotating turbulent flow, Physics of Fluids 33, 115128 (2021).
[2] Parfenyev, V. M., Vointsev, I. A., Skoba, A. O., & Vergeles, S. S. (2021). Velocity profiles of cyclones and anticyclones in a rotating turbulent flow. Physics of Fluids, 33(6), 065117.

We study the resonances in transmission of a subwavelength dielectric lossless structure, periodic in one direction and infinite in the orthogonal. We show that the diffraction type of the grid is determined by the geometric filling factor and the degree of asymmetry of the grid's profile. These two parameters are some linear functions of the complex amplitude of the second spatial Fourier harmonic of the material distribution in the grid. .
[1] Efremova, E. A., Perminov, S. V., & Vergeles, S. S. (2021). Resonance behavior of diffraction on encapsulated guided-mode grating of subwavelength thickness. Photonics and Nanostructures-Fundamentals and Applications, 46, 100953.

We construct a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov–Santini (MS) system. This map defines an Einstein–Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein–Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction in the MS system, which singles out the image of the dBKP equation solutions, and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein–Weyl structure for the BMS system.

We consistently develop a recently proposed scheme of matrix extensions of dispersionless integrable systems in the general case of multidimensional hierarchies, concentrating on the case of dimension d⩾4. We present extended Lax pairs, Lax–Sato equations, matrix equations on the background of vector fields, and the dressing scheme. Reductions, the construction of solutions, and connections to geometry are discussed. We separately consider the case of an Abelian extension, for which the Riemann–Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of the Penrose formula in curved space.

We give a short introduction to the integrable TT-deformation of QFT by Smirnov-Zamolodchikov. Basic properties of the deformed theories are discussed: factorization, Burgers equation for energy spectrum and exact deformed S-matrix. We study the lattice counterpart of the TT deformation for the case of RSOS(2, 2s+1) models. Starting from the deformed Bethe ansatz equations in thermodynamic limit, we obtain the energies and momenta of the ground and excited states, as well as the deformed breather S-matrix. In the scaling limit the results are in agreement with Smirnov-Zamolodchikov answers.

We consider smooth finite-parametric families of quasiperiodic potentials in the plane and the features of the Hamiltonian dynamics of particles in such potentials. As can be shown, the description of the geometry of the level lines of such potentials makes it possible to naturally divide such potentials into two classes, the first of which is in a sense closer to regular (periodic) potentials, and the second to random potentials. The geometry of the level lines of a potential should, certainly, be reflected in the features of the geometry of the trajectories when moving in such potentials, which really takes place in a certain energy interval. As shown by numerical studies, however, the dynamics of particles in the considered potentials has one more essential feature. The phase dynamic space is divided into areas in which the integrable dynamics takes place, and areas in which the dynamics has chaotic properties. The fraction of the regions corresponding to the integrable dynamics is large at low energy levels and decreases with increasing energy. In the region of nontrivial geometry of the potential level lines, both regimes are usually represented equally clearly, which entails the corresponding features in the description of the dynamics in such potentials. As an example, we present numerical results for families of potentials that often arise in the study of systems of cold atoms in a plane.