Theoretical and Mathematical Physics最新文献

In this paper, we classify the vacuum constraint equations of General Relativity using the representation theory of classical groups. We also provide general solutions for the differential equations of each class and prove vanishing (nonexistence) theorems for these solutions using the Bochner technique, which is an important tool in Geometric Analysis.

We consider compatibility conditions for equations describing nonuniform helical flows of an inviscid incompressible fluid. The system under study includes the Euler equations supplemented by differential constraints defining the helical flows. In the general case of an arbitrary function establishing relationships between the velocity and the vorticity vector, this system is not involutive. Since reducing this overdetermined system to involutive form generally leads to cumbersome calculations, the study focuses on functions identified as a result of preliminary group classification. This classification leads to several nonequivalent cases. We carry out a complete study of compatibility of the cases where the factor algebra modulo the kernel of the Lie algebra corresponding to the equivalence group has dimension greater than (2).

We consider a one-dimensional Bose gas model with attraction. We study the emptiness formation probability in ground state. We obtain an explicit expression for this probability in the case of large-volume system.

We obtain new results, as well as those complementing already known ones, concerning the construction of solutions of systems of differential equations corresponding to certain models of random matrix type. These solutions are expressed in terms of solutions of Painlevé II–V equations. We also show that solutions of systems of differential equations associated with random matrix type models having Laguerre and Hermitian kernels satisfy the formal Painlevé test. We obtain new formulas relating solutions of Painlevé III and Painlevé V equations under certain conditions imposed on the parameters entering these equations.

We present a review of publications devoted to exact solutions, transformations, symmetries, reductions, and applications of strongly nonlinear stationary and nonstationary (parabolic) equations of the Monge–Ampère type. We study the strongly nonlinear nonstationary mathematical physics equations with three independent variables that contain a quadratic combination of second spatial derivatives of the Monge–Ampère type and an arbitrary degree of the first temporal derivative or an arbitrary function depending on this derivative. We study the symmetries of these equations using group analysis methods. We derive formulas that enable the construction of multiparameter families of solutions, based on simpler solutions. We consider two-dimensional and one-dimensional symmetry and nonsymmetry reductions, which transform the original equations into simpler partial differential equations with two independent variables, or to ordinary differential equations and systems of such equations. Self-similar and other invariant solutions are described. Using generalized and functional separation of variables methods, we constructed several new exact solutions, many of which are expressed in elementary functions or in quadratures. Some solutions are obtained using auxiliary intermediate-point or contact transformations. These exact solutions can be used as test problems to verify the adequacy of and evaluate the accuracy of numerical and approximate analytical methods for solving problems described by strongly nonlinear mathematical physics equations.

A method for solving the double and triple sine-Gordon equations with first derivatives is presented. The search for solutions is similar to the search for functionally invariant solutions of the multidimensional wave equation. The solvability of the resulting system of equations is analyzed. The solution of the double sine-Gordon equation is obtained in explicit form by inverting the elliptic integral. The solution of the triple sine-Gordon equation requires inversion of the ultra-elliptic integral in the general case.

We study the problem with an unknown moving boundary of the conversion of (mathrm{CH}_4) hydrate into (mathrm{CO}_2) hydrate in a porous medium. We assume that in the initial state, methane hydrate coexists with water and free methane in the thermodynamic equilibrium state. Calculations show that the assumptions of the existence of the front conversion mode and of the constancy of saturations before the front in the mathematical model lead to the methane hydrate supercooling. We propose a generalized mathematical model that takes into account phase transitions in an extended region before the front. We find a self-similar solution of the problem in the linear approximation. Our made calculations show that the carbon dioxide injection with the conversion of methane hydrate into carbon dioxide hydrate is accompanied by the formation of methane hydrate before the front. We show that an amount of formed methane hydrate before the front increases with increasing injection pressure and permeability. We find that the hydrate formation in the mixture region increases the conversion front velocity.

We asymptotically obtain a generalized Schrödinger equation for nonlinear deformation waves in a metamaterial. It turns out to be analogous to the Sasa–Satsuma equation derived for optical waves. We study distinctions in the solution in the form of localized deformation waves related to the generalization of the Schrödinger equation.

As a result of the linearization of nonlinear equations for displacements in a nonlinear model of elastically conductive micropolar medium in a magnetic field on the background of a soliton solution describing subsonic solitary waves, we obtain an inhomogeneous scalar linear equation. This equation leads to a generalized spectral problem. To establish the instability of the mentioned solitary waves, the existence of an unstable eigenvalue (with a positive real part) must be verified. The corresponding proof is carried out by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right complex half-plane, and its zeros coincide with the unstable eigenvalues. It is proved that the Evans function tends to unity at infinity. This property of the Evans function, for some of its local properties in a neighborhood of the origin, allows us to conclude that it has zeros on the positive real semi-axis and therefore the subsonic solitary wave is unstable.