We discuss the mechanisms of unconventional super-conductivity and superfluidity in 3D and 2D fermionic systems with purely repulsive interaction at low densities.We construct phase diagrams of these systems and find the areas of the superconducting state in free space, as well as on the lattice in the framework of the Fermi-gas model with hard-core repul- sion, the Hubbard model, the Shubin±Vonsovsky model, and the t-J model. We demonstrate that the critical superconducting temperature can be greatly increased in the spin-polarized case or in a two-band situation already at low densities.

A Fermi gas described within the Bardeen-—Cooper—Schrieffer theory (BCS) may be converted into a Bose—Einstein condensate (BEC) of composite molecules (dimers) by adiabatically tuning the interaction. The sequence of the states that appears during this conversion is referred to as the BCS—BEC crossover. The review is devoted to theoretical and experimental results on the BCS—BEC crossover in three- and quasi-two-dimensional resonant quantum gases in the limiting geometry of traps and optical lattices. We shall discuss nontrivial phenomena in the superfluid hydrodynamics of the quantum gases and fluids including the spectrum of collective excitations in the BCS-BEC crossover, hydrodynamics of rotating Bose condensates with a large number of quantized vortices, and the complex unresolved problem of the chiral anomaly in the hydrodynamics of superfluid Fermi systems with anisotropic *p*-wave pairing. We shall also analyze spin-imbalanced quantum gases and the ability to realize the triplet *p*-wave pairing via the Kohn-Luttinger mechanism in these gases. The recent results on two-dimensional Fermi-gas preparation and observation of fluctuational phenomena related to the Berezinskii—Kosterlitz—Thouless transition in those gases will also be reviewed. In addition, we shall briefly discuss experimental realization of hexagon optical lattices with Dirac points in selected locations, which connects to the fast progress in the physics of mono- and bilayers of graphene and other Dirac semimetal systems. In addition we shall briefly discuss recently experimentally discovered BCS-BEC crossover and anomalous superconductivity in bilayer graphene and a possible role of graphene and 2D optical lattices as ideal systems for studying all the effects considered in this review.

Experimental and theoretical research on neutral excitations in a two-dimensional electron gas in a strong mag- netic field is reviewed. Methods for calculating excitation en- ergies in the strong-field limit for integer and noninteger filling factors are considered. The effects of impurities and of the nonideality of the two-dimensional system on the excitation spectrum are examined. Experimental results that have been obtained by the method of inelastic light scattering and that lend support to the current theoretical views are presented. We also discuss possible avenues of future experimental and theoretical work.

We discuss the properties of two-dimensional, non-

linear, potential, and vortex waves on the surface of an ideal

liquid of infinite depth. It is shown that in the quadratic order in

the amplitude, the vorticity of the Gerstner wave is equal in

magnitude to and different in sign from that of the Stokes drift

current in a surface layer. This allows a classic Stokes wave

obtained in the framework of potential theory to be interpreted

as a superposition of the Gerstner wave and Stokes drift. It is

proposed that the nonlinearity coefficient in the nonlinear

SchroÈ dinger equation can be physically interpreted as the Dop-

pler frequency shift along the vertically averaged Stokes drift

current.

This paper reviews the statistical properties and calculates the velocity structure functions of flows produced by a large-scale random scaling force in the Burgers model.

The system of thermals that makes the fine structure of a turbulent convective layer of a fluid is considered. A simplified probabilistic-geometrical approach is outlined that uses measurements along the observation line to determine the average in-plane parameters of the system. A dynamic equation for an isolated thermal interacting with its environment is derived. A Langevin equation similar to the stochastic equation for an ensemble of 'fast' Brownian particles is constructed for a system of thermals. The nonlinear Langevin equation for such a system leads to the associated kinetic form of the Fokker-Planck equation. It is shown that the stationary solution of the kinetic Fokker-Planck equation is identical to the Maxwell distribution and approximately consistent with the distributions measured in the turbulent convective layer of the atmosphere.

We outline the history and development of the theory of thin current sheets in a collisionless space plasma from the early ideas of V L Ginzburg and S I Syrovatskii to the present day. We review the key achievements of the quasi- adiabatic theory, which provided insight into the fine structure of thin current sheets and enabled a comparison with experi- ment. This comparison showed the quasi-adiabatic approach to be more effective than the classical MHD approximation. With the development of the quasi-adiabatic theory in the last two decades, the existence of a number of new thin current sheet features, such as multi-scaling, metastability, and em- bedding, has been predicted and subsequently confirmed in situ; the role of individual particle populations in the forma- tion of the current sheet fine structure has also been investi- gated. The role of nonadiabatic effects in accelerating plasma beamlets interacting with current sheets is examined. Asym- metry mechanisms in thin current sheets in the presence of a magnetic shear component are described. A study is carried out of current sheet self-organization processes leading to the formation of a shear magnetic component consistent with currents flowing in the plasma. It is demonstrated that the ongoing development of the theory of thin current structures is a logical continuation of Syrovatskii's and Ginzburg's ideas on cosmic rays and reconnected current sheets in the solar corona.

Earth's global magnetic field generated by an internal dynamo mechanism has been continuously changing on different time scales since its formation. Paleodata indicate that relatively long periods of evolutionary changes can be replaced by quick magnetic inversions. Based on observations, Earth's magnetic field is currently weakening and the magnetic poles are shifting, possibly indicating the beginning of the inversion process. This paper invokes Gauss coefficients to approximate the behavior of Earth's magnetic field components over the past 100 years. Using the extrapolation method, it is estimated that the magnetic dipole component will vanish by the year 3600 and at that time the geomagnetic field will be determined by a smaller value of a quadrupole magnetic component. A numerical model is constructed which allows evaluating and comparing both galactic and solar cosmic ray fluxes in Earth's magnetosphere and on its surface during periods of dipole or quadrupole domination. The role of the atmosphere in absorbing particles of cosmic rays is taken into account. An estimate of the radiation danger to humans is obtained for the ground level and for the International Space Station altitude of km. It is shown that in the most unfavorable, minimum field interval of the inversion process, the galactic cosmic ray flux increases by no more than a factor of three, implying that the radiation danger does not exceed the maximum permissible dose. Thus, the danger of magnetic inversion periods generally should not have fatal consequences for humans and nature as a whole, despite dramatically changing the structure of Earth's magnetosphere.

The unexpected discovery in 2006 of the first layered Fe-based high-temperature superconductor (HTSC), LnOFePn (where Ln is lanthanide, Pn is pnictide; hereafter referred to as 1111), becomes a key issue in modern solid-state physics. Since 2008, the class of iron-based superconductors has greatly expanded: several families of iron pnictides and chalcogen- ides have been synthesized. The crystal structure of oxypnictides is reminiscent of that of cuprates and is, in fact, a stack of superconducting Fe±As layers alternating along the c-direction with spacers, nonsuperconducting oxide blocks, LnO. In spite of the pronounced layered structure and anisotropic physical properties, the electron subsystem in Fe- based superconductors is less quasi-two-dimensional than that in cuprate HTSC, because the height of the Fe-As blocks exceeds the thickness of the CuO2 planes, whereas the distance between superconducting blocks in iron-based superconductors is significantly shorter than that in cuprates. This seems to be a reason why the critical temperature of Fe-based superconductors, though being as high as Tc=57K, still does not reach the cuprate one.

Influence of computational physics on the advanced recearch developmet discussed. Interesting examples of the application of computational physics to the testing of theory and hypotesis was presented.

Iron-based superconductors, just after their discovery in 2008, have become a subject of great interest for the scientific community and occupy one of the leading places among the most topical subjects in contemporary solid-state physics. The present development of investigations of iron-containing superconductors can be compared perhaps with the great efforts to study properties of cuprate high-temperature superconductors (HTSCs) in the first years after their discovery. At present more than one hundred Fe-based superconductors of different compositions have been found. These compounds represent quite a new class of superconducting materials with crystal lattice containing ions of 3d metals (Fe, Co, Ni) well known as ferromagnetic metals. Therefore, a priori, other mechanism of superconducting pairing in Fe-based superconductors, different from the traditional electron±phonon coupling, cannot be ruled out. A characteristic feature of all iron-containing superconduc- tors is the presence in their crystal structure of FeAs layers in the case of pnictides or FeSe layers in the case of chalcogenides. At present, the maximum critical temperature Tc of the superconducting transition in Fe-based super- conductors reaches 56 K (in a Gd1-xThxFeAsO compound), which is inferior only to Tc of cuprate HTSCs. This circumstance undoubtedly makes it possible to place ironbased superconductors in the class of HTSCs.

A study is made of the spinon continuum fine structure which is observed to occur in the spin-liquid phases of chain- and triangular-lattice magnets at small wave vectors due to the action of a uniform Dzyaloshinsky Moriya interaction. An ordered phase with a strongly quantum-reduced order parameter is found to exhibit the coexistence of magnon and spin type excitations, the former crossing over to the latter when the excitation energy exceeds that of the exchange interaction.

Some important results of the 70 years of theoretical research at the Alikhanov Institute for Theoretical and Experimental Physics (ITEP) are reviewed.

Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors can be realized, analogous to those occurring in dissipative systems, like stable equilibrium points, corresponding to stationary rotations, limit cycles (rotations with oscillations), strange attractors. A chart of dynamical regimes in the parameter plane of a total mechanical energy and a relative angle between the geometric and dynamic principal axes of the body is plotted and discussed. The transition to chaos through a sequence of Feigenbaum period doubling bifurcations is demonstrated. Several examples of strange attractors are considered; their phase portraits, Lyapunov exponents, and Fourier spectra are discussed.

The sum-frequency generation involving two infrared laser quanta and a single visible-range laser quantum is a four-wave mixing process that is virtually not used in practice. Nevertheless, this process provides an extremely high selectivity with respect to the Q-branch of the two-photon vibrational transition in molecules. We explore here two publications: one that is more than thirty years old, and another that appeared in 2018, to show broad potential applications of the method. The objective reasons why this potential has not been used so far are discussed.