Theoretical and Mathematical Physics最新文献

We study an axion soliton that weakly interacts with background matter and magnetic fields. A mirror-symmetric soliton whose magnetic flow is due to secondary magnetic helicity invariant is described by the Iroshnikov–Kraichnan spectrum. For a large-scale magnetic field, a dynamo is not observed. In a mirror axionic soliton, a phase transition producing a magnetic helical flow is possible. Using this transition, the soliton becomes mirror-asymmetric. When the mirror symmetry is broken, the axion soliton acquires magnetic energy, which is the result of the transformation of the axionic energy. Our main result for the initial stage of the process is calculating a scale for which the generation of large-scale magnetic fields is the most intense. Numerical simulations show that lower lateral harmonics of the magnetic field have smaller amplitudes compared to higher ones. We study the simplest statistical ensemble defined by the projection of all harmonics onto principal ones. We conjecture that a certain instability in axionic MHD is observed. We propose a possible explanation for this phenomenon. When the mirror symmetry of the axion soliton is broken, the (gamma)-term in the axionic mean-field equation, which is related to the axion spatial inhomogeneity, interacts with principal harmonics. As a result, the axion soliton acquires magnetic energy and becomes helical.

We construct a Hamiltonian formalism of general relativity theory, where the worldsheet metric, the metric of the three-dimensional space, and their mutual influence are explicitly singled out. We choose a gauge condition corresponding to a nonstationary background solution. In the quadratic approximation, we study the stability of the background solution and define particles as excitations over this solution.

We construct the generalized Darboux transformation for the modified complex short pulse (mcSP) equation. As applications, we present high-order solutions for the mcSP equation by the generalized Darboux transformation. In the case of zero seed solution, high-order soliton solutions are derived, and in the case of plane wave seed solution, high-order rogue wave solutions are obtained. As examples, some high-order solutions and their dynamics are illustrated graphically.

We study the propagation of longitudinal waves in a homogeneous, nonlinearly elastic rod located in an external nonstationary magnetic field, in the presence of damage in the rod material. The dynamic behavior of the rod is determined by Bishop’s theory. We consider the initial system of equations in two limiting cases and in the general case. In the first limiting case, we assume that under conditions of a strong magnetic field, the rod material has a high electrical resistance. In the second limiting case, we assume that the rod material has the property of ideal conductivity. In each particular case, the system reduces to a single nonlinear fifth-order equation for the longitudinal displacement of rod particles. Taking the short relaxation time into account, we obtain evolution equations for the longitudinal deformation function representing the well-known wave dynamics equation—the Kuramoto–Sivashinsky equation and its generalization containing an additional quadratically nonlinear term. We find exact solutions of the obtained evolution equations using the simplest equations method. We show that the solutions describe spatially localized deformation waves in the form of solitons and shock waves. We analyze the dependences of the characteristic parameters of stationary waves (amplitude, front width, and propagation velocity) on the system parameters. In the general case, the system reduces to a nonlinear seventh-order equation. In ordinary derivatives and under certain relations between the parameters, the equation transforms into an anharmonic oscillator equation with two types of quadratic nonlinearity. We find the first integral of the equation. The performed qualitative analysis shows the possibility of propagation of deformation waves in the system: nonlinear periodic and spatially localized soliton-type waves.

We calculate the amplitude of the process of generating a pair of a massive neutrino and antineutrino, to which a third-rank tensor caused by the contribution of an electron “three-pole” with two vector vertices and one “weak” vertex corresponds. We analyze procedures for reducing logarithmic divergences when integrating over the loop momentum and gauge-noninvariant contributions, obtaining a final gauge-invariant and finite result. We find the cross section for this inelastic process.

We discuss an aspect of the application of the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear differential equations. The aspect is related to the number of balance equations needed to obtain an exact solution of nonlinear differential equations solved. The work and results of Professor Kudryashov stimulated our research on SEsM at an important period in the development of this methodology. Because of this, we start with a short description of SEsM and then briefly review our research on the exact solution of nonlinear differential equations, as well as some of the results of Prof. Kudryashov in this area in the last 30 years. We apply the specific case (mathrm{SEsM}(1,1)) of the SEsM to the following class of nonlinear differential equations:

where (A_{f,omega,omega_1}bigl(F,bigl{frac{partial^{zeta}F}{partial x^{zeta_1}partial t^{zeta-zeta_1}}bigr}bigr)) and (B(F)) are polynomials in the unknown function (F) and its derivatives. As a simple equation, we use an ordinary differential equation (bigl(frac{dPhi}{dxi}bigr)^epsilon=sum_{pi=0}^{sigma}gamma_{pi}[Phi (xi)]^pi), which contains as a specific case, the elliptic equation (bigl(frac{dPhi}{dxi}bigr)^2=aPhi^4+bPhi^2+c). We show that this can lead to the necessity of using more than one balance equation. The methodological results are illustrated by selected simple examples.

We define families of Kuramoto models related to bounded symmetric domains. The families include Lohe unitary and spherical models as special cases. Our approach is based on the construction proposed by Watanabe and Strogats. We replace the Poincare disc and its (S^1) boundary with bounded symmetric domains and with its Bergman–Shilov boundaries. In Cartan’s classifications there are four classical domains of types I–IV. Here we consider the domains of types I, II, and III. For a fixed domain, there is a decreasing chain of components of Bergman–Shilov boundaries. This leads to the families of Kuramoto models that we describe here.

We survey results on the structural stability of shock waves in elastodynamics of compressible neo-Hookean materials. By nonlinear structural stability of a shock wave we mean the local-in-time existence and uniqueness of the discontinuous shock front solution of the elastodynamics equations, which guarantees the real existence of the shock wave as a physical structure. We describe finding structural stability conditions for shock waves in 2D elastodynamics using both the energy method and spectral analysis of the corresponding linearized free boundary problem. We also briefly discuss recent results on structural stability in the general 3D case.

We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity (|D_xu|^q). Two exponents, (q_1=N/(N-1)) and (q_2=(N+1)/(N-1)), are found such that for (1<qleq q_1), there is no weak solution local in time for any (T>0); for (q_1<qleq q_2), there is a unique weak solution local in time; however, there is no weak solution global in time, i.e., independently of the “value” of the initial function, the solution to the Cauchy problem blows up in a finite time.

We study the linear stability of the vertical flow that occurs when gas displaces oil from a layer of porous medium using the generalized nonlinear Forchheimer filtration law. We consider the case where areas saturated with oil and gas are separated by a layer of water. The interfaces separating the areas are assumed to be flat at the initial moment. We consider two cases of perturbation evolution. In the first case, only the gas–water interface is perturbed at the initial moment. In the second case, small perturbations of the same amplitude are present on both surfaces. We show that the interaction of perturbations at interfaces depends on the thickness of the water-saturated layer, perturbation wavelength, oil viscosity, pressure gradient, and formation thickness. Calculations demonstrate that perturbations at the oil–water boundary grow much slower than perturbations at the gas–water boundary. We find that there is a critical value of the thickness of the water-saturated layer. If the thickness of the layer is greater than the critical value, then the development of perturbations at the gas–water boundary does not affect the development of perturbations at the water–oil boundary.