Transmission of microwaves through a composite plate containing Fe nanoparticles in an epoxyamine matrix, as well as reflection of waves from it, has been investigated. The experiments were performed at the frequencies from 26 to 38 GHz in the magnetic fields up to 12 kOe. The ferromagnetic resonance line in the composites with the weight fraction of Fe particles from 10% to 30% has been studied. The magnetic field dependence of the microwave power dissipation has been plotted. Field dependence of the transmission and reflection coefficients have been calculated, as well as qualitative, and in some cases quantitative, agreement has been obtained. The penetration depth of microwaves into the composites has been analyzed. Spectrum of the FMR has been constructed. Results of interaction of microwaves with Fe nanoparticles are discussed taking into account magnetic properties and composite structure.

The propagation of microwaves through the dielectric substrate/metal superlattice system [(CoFe)/Cu] has been investigated. The frequency dependences of transmission and reflection coefficients for the normal incidence of electromagnetic waves on the system in two opposite directions are measured. The effect of the substrate thickness on the magnitude of the microwave giant magnetoresistance effect during reflection and on nonreciprocity in the system is investigated. The influence of an external magnetic field on the nonreciprocity parameter is investigated. It has been established that under the conditions of nonreciprocity, the microwave giant magnetoresistance effect increases significantly when the wave is reflected. Keywords: metal superlattices, microwave giant magnetoresistance effect, microwaves, transmission and reflection coefficients, nonreciprocity.

Microwave transmission through plates of a composite material containing spherical Fe nanoparticles epoxyamine matrix and reflection from plates have been studied. Measurements were carried out quencies from 26 to 32 GHz in the magnetic fields up to 12 kOe. The ferromagnetic resonance phenomenon composite has been investigated. The theory of electromagnetic waves transmitting through a composite containing ferromagnetic particles, taking into account aggregating the particles, has been developed. agreement of calculation results and experimentally obtained field dependences of the transmission reflection coefficients, as well as microwaves dissipation, has been achieved.

The microwave magnetoresistance of CoFe/Cu/CoFe/FeMn spin valves and CoFe/Cu/CoFe three-layer nanostructures with high magnetoresistance has been studied. The transmission and reflection coefficients were measured at the frequency range from 26 to 38 GHz in magnetic fields up to 12 kOe. It is shown that the dependences of the transmission coefficient of spin valves are not symmetric with respect to the H=0 axis, as well as the dependences of magnetoresistance. It is established that the relative changes in the microwave transmission coefficient are 1.5-2 times higher than the relative magnetoresistance measured at direct current. Changes in the reflection coefficient have a smaller value and the opposite sign with respect to changes in the transmission coefficient. Keywords: metal superlattices, spin valves, ferromagnetic resonance, ferromagnetic antiresonance, microwave giant magnetoresistance effect.

The paper presents a brief review of the quasiclassical wave dynamics for the nonlinear Schrödinger equation (NLSE) as applied to focusing and defocusing media. The NLSE depends significantly on the space dimension d. The two-dimensional NLSE has an additional symmetry of the conformal type with respect to the Talanov transformations (Talanov in JETP Lett. 11:199–201, 1970), which were initially found for the stationary self-focusing in a medium with the Kerr nonlinearity. A consequence of this symmetry is the Vlasov–Petrishchev–Talanov theorem (Vlasov et al. in Radiophys. Quantum Electron. 14:1062–1070, 1971) that relates the mean of the squared distribution and the Hamiltonian of the system. This theorem is valid for both focusing and defocusing media. In the quasiclassical limit, this makes it possible to construct anisotropic solutions which describe beam compression during self-focusing and quantum-gas expansion into vacuum within the so-called critical nonlinear Schrödinger equations, in particular, for the Gross–Pitaevskii equation with a chemical potential having a power-law dependence on density with the exponent ν = 2/d. For the Gross–Pitaevskii equation, the case d = 2 corresponds to a condensate of a weakly nonideal Bose gas, and the case d = 3 describe condensate of a Fermi gas in the unitary limit. For d = 3, the Gross–Pitaevskii equation in the quasiclassical limit transforms into equations of the gas dynamics with the adiabatic exponent γ = 5/3. The self-similar solutions in this approximation describe the angular deformations of a gas cloud against the background of an expanding gas. Angular deformations of such type are observed in both the expansion of quantum gases and the action of high-power laser radiation on matter. For three-dimensional supercritical focusing NLSE, the quasiclassical solutions of the collapsing type are presented, including the exact semiclassical solution described by the strong collapse regime. It is found that all such quasiclassical collapses are found to be unstable, except for the collapse that is simultaneously the weakest and the fastest collapse corresponding to the self-similar NLSE solution. The problem of post-collapse is also considered as the continuation of a weak collapse, which results in the formation of a quasistationary singularity in the form of a black hole into which energy is drawn from the surrounding collapsing region. For the NLSE with d ≥ 4, the formation of a black hole can be described in the quasiclassical approximation. It is shown that the anisotropy caused by the magnetic field significantly alters the structure of the Langmuir collapse, in particular, leads to the formation of strongly anisotropic black holes described quasiclassically.

The microwave giant magnetoresistance effect in a (CoFe/Cu) superlattice with micron-sized holes has been studied. Measurements of the frequency dependences of the transmission coefficient, as well as the dependences of the microwave transmission and reflection coefficients on the magnetic field, are performed. The measurements were performed on the superlattice samples without holes, having one hole with a diameter of 6.3 μm and seven holes with a diameter of 1.7 μm. It is shown that the presence of a hole with a diameter of 6.3 μm leads to a significant frequency dependence of the microwave giant magnetoresistance effect. Magnetic and magnetoresistance measurements of superlattice samples were performed. Keywords: metal superlattices, ferromagnetic resonance, ferromagnetic antiresonance, microwave giant magnetoresistance effect.

Experiments on the giant magnetoresistance effect observed in microwave reflection have been conducted in the frequency range of 26–38 GHz on (CoFe)/Cu superlattices exhibiting giant magnetoresistance. The amount of the effect has been determined (up to +3% maximum), and its magnetic field dependence has been found. The microwave reflection coefficient versus magnetic field dependence has been calculated. The measured variation of the reflection coefficient has turned out to be greater than calculated values. This discrepancy is associated with the fact that a superlattice is approximated by a homogeneous plate in calculations. The frequency dependence of the giant magnetoresistance effect in microwave reflection has been observed. It is explained by the influence of the impedance of a waveguide in which samples are placed to take measurements.

Reply to the comment of P.L. Chapovsky to the paper “Symmetry approach in the problem of gas expansion into vacuum,” JETP 159, 794 (2021).

We develop a new concept for the formation of behavior features of inviscid incompressible fluids on the rigid boundary due to breaking of slipping flows. The breaking possibility is related to the compressibility of such flows due to the boundary. For two- and three-dimensional inviscid Prandtl equations, we analytically obtain the criteria for a gradient catastrophe for slipping flows. For the two-dimensional Prandtl equations, breaking occurs for both the velocity component parallel to the boundary and the vorticity gradient. The explosive growth of the vorticity gradient correlates with the appearance of a jet in the direction perpendicular to the boundary. For the three-dimensional Prandtl flows, breaking (fold formation) leads to an explosive growth for both the symmetric part of the velocity-gradient tensor and its antisymmetric part, i.e., vorticity. The blow-up generation of vorticity is possible due to the fluid suction from the slipping flow with simultaneous formation of a jet perpendicular to the boundary. These factors can be considered as a tornado-formation mechanism. Within the framework of the two-dimensional Euler equations, we numerically study the problem of the formation of increasing velocity gradients for the flows between two parallel plates. It is revealed that on the rigid boundary, the maximum velocity gradient exponentially increases with time simultaneously with an increase in the vorticity gradient according to the double exponential law. This process is also accompanied by a jet formation in the direction perpendicular to the boundary.

The anisotropy due to a magnetic field is shown to result in significant changes in Langmuir collapse. Using a variational approach, the quasi-classical collapse phenomenon is investigated analytically. A hierarchy of quasi-classical collapses is determined, along with the structure of a field in the proximity of a stationary singularity that is sustained by the continuous absorption of wave energy from a wave packet.

Reply to the comment of P.L. Chapovsky to the paper “Symmetry approach in the problem of gas expansion into vacuum,” JETP 132 , 704 (2021).

The studies of a number of systems treated in terms of an inhomogeneous (spatially separated) Fermi-Bose mixture with superconducting clusters or droplets of the order parameter in a host medium with unpaired normal states are reviewed. A spatially separated Fermi-Bose mixture is relevant to superconducting Ba-KBiO3 bismuth oxides. Droplets of the order parameter can occur in thin films of a dirty metal, described in the framework of the strongly attractive two-dimensional Hubbard model at a low electron density with a clearly pronounced diagonal disorder. The Bose-Einstein condensate droplets are formed in mixtures and dipole gases with an imbalance in the densities of the Fermi and Bose components. The Bose-Einstein condensate clusters also arise at the center or at the periphery of a magnetic trap involving spin-polarized Fermi gases. Exciton and plasmon collapsing droplets can emerge in the presence of the exciton-exciton or plasmon-plasmon interaction. The plasmon contribution to the charge screening in MgB2 leads to the formation of spatially modulated inhomogeneous structures. In metallic hydrogen and metal hydrides, droplets can be formed in shock-wave experiments at the boundary of the first-order phase transition between the metallic and molecular phases. In a spatially separated Fermi-Bose mixture arising in an Aharonov-Bohm interference ring with a superconducting bridge in a topologically nontrivial state, additional Fano resonances may appear and collapse due to the presence of edge Majorana modes in the system.

We present the results of direct numerical simulation of three-dimensional acoustic turbulence in medium with weak positive dispersion. It is shown that at the beginning of the long-wavelength region in the turbulence energy distribution in the k -space, there are formed jets in the form of narrow cones. At higher wavenumbers, the cones broaden, and the distribution accordingly tends to isotropic. In this region of wavenumbers, the angle-averaged turbulence spectrum acquires a power-law character, E(k) ∝k^ - α , with the exponent close to 3/2, which corresponds to the Zakharov–Sagdeev weak acoustic turbulence spectrum.

This article is a brief review of the results of studying the collapse of sound waves in media with positive dispersion, which is described in terms of the three-dimensional Kadomtsev–Petviashvili (KP) equation. The KP instability of one-dimensional solitons in the long-wavelength limit is considered using the expansion for the corresponding spectral problem. It is shown that the KP instability also takes place for two-dimensional solitons in the framework of the three-dimensional KP equation with positive dispersion. According to Kadomtsev (see Collective Effects in Plasma by B.B. Kadomtsev) this instability belongs to the self-focusing type. The nonlinear stage of this instability is a collapse. One of the collapse criteria is the Hamiltonian unboundedness from below for a fixed momentum projection coinciding with the L2-norm. This fact follows from scaling transformations, leaving this norm constant. For this reason, collapse can be represented as the process of falling a particle to the center in a self-consistent unbounded potential. It is shown that the radiation of waves from a region with a negative Hamiltonian, due to its unboundedness from below, promotes the collapse of the waves. This scenario was confirmed by numerical experiments. Two analytical approaches to the study of collapse are presented: using the variational method and the quasiclassical approximation. In contrast to the nonlinear Schrödinger equation (NLSE) with a focusing nonlinearity, a feature of the quasiclassical approach to describing acoustic collapse is that this method is proposed for the three-dimensional KP equation as a system with hydrodynamic nonlinearity. Within the framework of the quasiclassical description, a family of self-similar collapses is found. The upper bound of this family corresponds to a strong collapse, in which the energy captures into the singularity is finite. The existence of such a regime is also confirmed based on the variational approach. The other boundary of the collapsing hierarchy coincides with the self-similar solution of the three-dimensional KP equation, which describes the fastest weak collapse.

We study the formation of quasi-two-dimensional (thin pancake) vortex structures in three-dimensional flows and of quasi-one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at large Reynolds numbers, when their evolution is described in the leading order by the Euler equations for an ideal incompressible fluid. We show numerically and analytically that the compression of these structures and, as a consequence, the increase in their amplitudes are due to the compressibility of the frozen-in-fluid fields: the field of continuously distributed vortex lines in the three-dimensional case and the field of vorticity rotor lines (divorticity) for two-dimensional flows. We find that the growth of vorticity and divorticity can be considered to be a process of overturning the corresponding fields. At high intensities, this process demonstrates a Kolmogorov-type scaling relating the maximum amplitude to the corresponding thicknesses-to-width ratio of the structures. The possible role of these coherent structures in the formation of the Kolmogorov turbulent spectrum, as well as in the Kraichnan spectrum corresponding to a constant flux of enstrophy in the case of two-dimensional turbulence, is analyzed.

The microwave magnetoresistance of CoFe/Cu/CoFe/FeMn spin valves and CoFe/Cu/CoFe three-layer nanostructures with high magnetoresistance has been studied. The transmission and reflection coefficients were measured at the frequency range from 26 to 38 GHz in magnetic fields up to 12 kOe. It is shown that the dependences of the transmission coefficient of spin valves are not symmetric with respect to the H=0 axis, as well as the dependences of magnetoresistance. It is established that the relative changes in the microwave transmission coefficient are 1.5-2 times higher than the relative magnetoresistance measured at direct current. Changes in the reflection coefficient have a smaller value and the opposite sign with respect to changes in the transmission coefficient. Keywords: metal superlattices, spin valves, ferromagnetic resonance, ferromagnetic antiresonance, microwave giant magnetoresistance effect.

Ferromagnetic-resonance-induced variations in the transmittance of Fe films and Fe/Cr superlattices are studied in a microwave frequency interval of 26–38 GHz. The shape of the resonance line is described using a model in which the asymmetry is provided by a Lorentzian dispersion curve added to the absorption curve. It is shown that the line shape is well described using the model for superlattices with continuous Fe and Cr layers and Fe films. However, only qualitative agreement is obtained for superlattices with thin Fe and Cr layers. The experimental field dependence of the transmission coefficient substantially differs from the model results in the presence of the fields that are less than the field of ferromagnetic resonance for superlattices with giant magnetoresistance.

The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows are compressible, which is the main reason for the formation of a singularity for the gradient of the velocity component parallel to rigid border. Slipping flows are studied analytically in the framework of two- and three-dimensional inviscid Prandtl equations. Criteria for a gradient catastrophe are found in both cases. For 2D Prandtl equations breaking takes place both for the parallel velocity along the boundary and for the vorticity gradient. For three-dimensional Prandtl flows, breaking, i.e. the formation of a fold in a finite time, occurs for the symmetric part of the velocity gradient tensor, as well as for the antisymmetric part - vorticity. The problem of the formation of velocity gradients for flows between two parallel plates is studied numerically in the framework of two-dimensional Euler equations. It is shown that the maximum velocity gradient grows exponentially with time on a rigid boundary with a simultaneous increase in the vorticity gradient according to a double exponential law. Careful analysis shows that this process is nothing more than the folding, with a power-law relationship between the maximum velocity gradient and its width: $% \max|u_x|\propto \ell^{-2/3}$.