We investigate the conductance of a 1D disordered conducting loop with two contacts, immersed in a magnetic flux. We show the appearance in this model of the Al'tshuler-Aronov-Spivak behaviour. We also investigate the case of a chain of loops distributed with finite density: in this case we show that the interference effects due to the presence of the loops can lead to the delocalization of the wave function.

Recently discussed topological materials Weyl-semimetals (WSs) combine both: high electron mobility comparable with graphene and unique topological protection of Dirac points. We present novel results related to electromagnetic field propagation through WSs. It is predicted that transmission of the normally incident polarized electromagnetic wave (EMW) through the magnetic WS strongly depends on the orientation of polarization with respect to a gyration vector g. The latter is related to the vector-parameter b, which represents the separation between the Weyl nodes of opposite chirality in the first Brillouin zone. By changing the polarization of the incident EMW with respect to the gyration vector g the system undergoes the transition from the isotropic dielectric to the medium with Kerr-or Faraday-like rotation of polarization and finally to the system with chiral selective electromagnetic field. It is shown that WSs can be applied as the polarization filters.

We consider three Ginibre ensembles (real, complex and quaternion-real) with deformed measures and relate them to known integrable systems by presenting partition functions of these ensembles in form of fermionic expectation values. We also introduce double deformed Dyson–Wigner ensembles and compare their fermionic representations with those of Ginibre ensembles.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

By employing a simple model for small-scale linear edge waves propagating along a homogeneous sloping beach, we demonstrate that certain combinations of linear wave components may lead to durable changes in the thickness of the surfactant film, equivalently, in the concentration of various substances (debris, litter) floating on the water surface. Such changes are caused by high-amplitude transient elevations that resemble rogue waves and occur during dispersive focusing of wave fields with a continuous spectrum. This process can be treated as an intrinsic mechanism of production of patches in the surface layer of an otherwise homogeneous coastal environment impacted by linear edge waves.

We describe a new functional integral method for the computation of averages containing chronological exponentials of random matrices of arbitrary dimension. We apply these results to the rigorous study of the statistics of a passive scalar advected by a large-scale *N*-dimensional flow. In the delta-correlated case the statistics of the rate of line stretching appears to be exactly Gaussian at all times and we explicitly compute the dependence of the mean value and variance of the stretching rate on the space dimension *N*. The probability distribution function of the passive scalar is also exactly computed. Further applications of our functional integral method are suggested.

The partition function of the quantum Heisenberg ferromagnet is represented in a closed functional form. The resulting expression is a path integral for two interacting scalar Bose fields with non-polynomial action.

Fine features of gamma-ray radiation registered during a thunderstorm at Tien-Shan Mountain Cosmic Ray Station are presented. Long duration (100–600 ms) gamma-ray bursts are found. They are for the first time identified with atmospheric discharges (lighting). Gamma-ray emission lasts all the time of the discharge and is extremely non-uniform consisting of numerous flashes. Its peak intensity in the flashes exceeds the gamma-ray background up to two orders of magnitude. Exclusively strong altitude dependence of gamma radiation is found. The observation of gamma radiation at the height 4–8 km could serve as a new important method of atmospheric discharge processes investigation.

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting

Results of experiments demonstrating the phenomenon of runaway electron breakdown of atmospheric air under laboratory conditions are presented. As the discharge-initiating electron beam of duration ∼50 ps∼50 ps had passed through the electrode gap, a runaway electron avalanche current was detected in the electrode gap downstream of the anode grid and then breakdown occurred with picosecond stability. The maximum electron energy and the duration of the avalanche current corresponded to theoretical notions about the runaway electron breakdown of atmospheric air in a strong electric field. Breakdown did not occur at all or was considerably delayed when no initiating beam was used.

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.

The results are presented of the first investigation of linear and nonlinear processes associated with waves which are related to the presence of magnetic fields in dusty plasmas at the Moon. Excitation of lower-hybrid turbulence in dusty plasmas near the lunar surface is studied. It is shown that the lower-hybrid turbulence can be generated wherever the Earth's magnetotail interacts with the near-surface dusty plasmas at the Moon. The electric fields appearing as a consequence of the presence of lower-hybrid turbulence are estimated. They can make a significant contribution to the total electric field above the lunar surface which should be taken into account in the future experimental investigation of electric fields at the Moon.

The Lagrangian evolution of material elements has been extensively studied in various works [1], [2] and [3]. These studies were connected with different problems: such as Lagrangian turbulence [5] or passive scalar decay [2]. One of the most important results of these studies is the proof of the intermittency in structure functions. This intermittency seems closely connected with the intermittency of the developed turbulence

A new method of the mean field type is proposed for the density of states calculation in the gaussian random potential with gaussian correlator. This method becomes more precise when the dimension of space increases.

Formation of exciton-polariton condensate due to incoherent pumping of an excitonic reservoir is considered. The condensate dynamics is governed by a system of stochastic integro-differential equations of Langevin's type corresponding to the model developed by Elistratov and Lozovik [12]. Attention is concentrated on non-Markovian interaction of the condensate with the excitonic and photonic reservoirs. It is shown that dynamical memory caused by the non-Markovian interaction qualitatively changes the condensate behavior as compared to the Markovian regime. In particular, it diminishes the threshold of pumping strength for the condensate emergence. Also, it is found that the non-Markovian regime leads to relaxation oscillations corresponding to population exchange between the condensate and the excitonic reservoir. Increasing of incoherent pumping strength leads to chaos of relaxation oscillations that is accompanied by random-like transitions between the lower and upper polaritonic states.

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C^2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing–van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing–van der Pol system are constructed.

We present a bi-Hamiltonian structure for the two-component Novikov equation. We also show that proper reduction of this bi-Hamiltonian structure leads to the Hamiltonian operators found by Hone and Wang for the Novikov equation.